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ELEMENTS 

OF 

MACHINE DESIGN 






VMe Qraw-JJillBook (a 7m 

PUBLISHERS OF BOOKS FOR./ 

Coal Age v Electric Railway Journal 
Electrical World v Engineering News -Record 
Railway Age Gazette v American Machinist 
Electrical Merchandising ^ The Contractor 
Engineering 3 Mining Journal ^ Po v/e r 
Metallurgical & Chemical Engineering 



ELEMENTS 

OF 

MACHINE DESIGN 



BY 
O. A. LEUTWILER, M. E. 

PROFESSOR OF MACHINE DESIGN, UNIVERSITY OF ILLINOIS 

MEMBER OF THE AMERICAN SOCIETY OF 

MECHANICAL ENGINEERS 



First Edition 



McGRAW-HILL BOOK COMPANY, Inc. 

239 WEST 39TH STREET. NEW YORK 



LONDON: HILL PUBLISHING CO., Ltd. 

6 & 8 BOUVERIE ST., E. C. 

1917 



<^ 



V 



\ 



Copyright, 1917, by the 
McGraw-Hill Book Company, Inc. 



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$ 



v 



OCT? 1917 



LE PRESS YORK PA 



A4'73812 
>£-l9 / 



PREFACE 

The purpose of the author, in preparing this book, has been 
to present in fairly complete form a discussion of the fundamental 
principles involved in the design and operation of machinery. An 
attempt is also made to suggest or outline methods of reasoning 
that may prove helpful in the design of various machine parts. 
The book is primarily intended to be helpful in the courses of 
machine design as taught in the American technical schools and 
colleges, and it is also hoped that it may prove of service to the 
designers in engineering offices. 

Since a text on machine design presupposes a knowledge of 
Strength of Materials and Mechanics of Machinery, a chapter 
reviewing briefly the more important straining actions to which 
machine parts are subjected is included as well as a chapter dis- 
cussing briefly the properties of the common materials used in the 
construction of machinery. Furthermore, throughout the book, 
the question of the application of mechanical principles to ma- 
chines and devices has not been overlooked, and many recent 
devices of merit are illustrated, described and analyzed. A 
considerable amount of the material in this book was published 
several years ago in the form of notes which served as a text in the 
courses of machine design at the University of Illinois. 

In the preparation of the manuscript the author consulted 
rather freely the standard works on the subject of machine 
design, the transactions of the various national engineering socie- 
ties and the technical press of America and England. Whenever 
any material from such sources of ' information was used, the 
author endeavored to give suitable acknowledgment. The nu- 
merous illustrations used throughout the book have been selected 
with considerable care and in the majority of cases they represent 
correctly to scale the latest practice in the design of the parts of 
modern machines. At the close of nearly every chapter a brief 
list of references to sources of additional information is given. 

Through the generosity '.at various manufacturers, drawings 
illustrating the prevailing practice in America were placed at the 
author's disposal thus making it possible to use scale drawings for 



vi PREFACE 

illustrating the various machine parts. To all such manufac- 
turers he is especially indebted. The author is also indebted to 
Mr. H. W. Waterfall of the College of Engineering of the Univer- 
sity of Illinois for the many helpful suggestions and criticisms 
received during the preparation of the manuscript. To his friend 
and colleague Professor G. A. Goodenough, the author is deeply 
indebted for much valuable advice and the many suggestions 
received in preparing the manuscript, also for the critical reading 
of the proof. 

O. A. L. 

Urbana, Illinois, 
September, 1917. 



CONTENTS 

CHAPTER I 

Page 

Stresses and Strains in Machine Parts 1 

Forces — Principles Governing Design — Stress and Strain — Stress- 
strain Diagram — Modulus of Elasticity — Poisson's Ratio — Resili- 
ence — Tensile Stress — Compressive Stress — Shearing Stress — 
Stresses Due to Flexure — Flexure Combined with Direct Stress — 
Straight Prismatic Bar— Offset Connecting Link — Stresses in Col- 
umns — Shearing Combined with Tension or Compression — Stresses 
Due to Suddenly Applied Forces — Repeated High Stresses — 
Repeated Low Stresses — Safe Endurance Stress — Deformation Due 
to Temperature Change — Stresses Due to Temperature Change — 
Factor of Safety — References. 

CHAPTER II 

Materials Used in the Construction of Machine Parts 26 

Cast Iron — Vanadium Cast Iron — Pig Iron — Malleable Casting — 
Chilled Casting — Semi-steel — Wrought Iron — Manufacturing 
Processes — -Manganese-steel Castings — Applications of Manga- 
nese-steel Castings — Bessemer Process — Open-hearth Process — 
Cementation Process— Crucible Steel — Cold-rolled Steel — Nickel 
Steel — Chrome Steel — Vanadium Steel — Nickel-chromium Steel — 
Chromium-vanadium Steel — Silicon-manganese Steel — Tungsten 
Steel — Brass— Bronze— Monel Metal — Aluminum — Babbitt Metal 
— Heat-treating Processes — S. A. E. Heat Treatments — Galvanizing 
— Shererdizing. 

CHAPTER III 

Fastenings — Rivets and Riveted Joints 48 

Rivets — Rivet Holes — Forms of Rivets — Forms of Heads — 
Types of Joints — Failure of Joints— Definitions — Analysis of a 
Boiler Joint — Efficiency of the Joint — Allowable Stresses — Mini- 
mum Plate Thickness — Design of a Boiler Joint — Rivet Spacing in 
Structural Joints — Types of Structural Joints — Single Angle and 
Plate — End Connections for Beams — Double Angle and Plate — 
Splice Joint — Pin Plates — Diagonal Boiler Brace — References. 

CHAPTER IV 

Fastenings — Bolts, Nuts, and Screws 76 

Forms of Threads — Bolts — Screws — Stay Bolts— Nut Locks — 
Washers — Efficiency of V Threads — Stresses Due to Screwing Up — 
Stresses Due to the External Forces — Stresses Due to Combined 
Loads — Fastening with Eccentric Loading — Common Bearing- 
Efficiency of Square Threads — Stresses in Power Screws. 



Vlil CONTENTS 

CHAPTER V 

Pag* 

Fastenings — Keys, Cotters, and Pins 110 

Sunk Keys — Keys on Flats — Friction Keys — The Strength of Keys 
— Friction of Feather Keys — Gib-head Keys — Key Dimensions — 
Integral Shaft Splines — Analysis of a Cotter Joint — Taper Pins — 
Rod and Yoke Ends — References. 

CHAPTER VI 

Cylinders Plates and Springs 129 

Thin Cylinders — Thick Cylinders — Rectangular Plates — Square 
Plates — Circular Plates — Flat Heads of Cylinders — Elliptical 
Plates — Helical Springs — Concentric Helical Springs — Helical 
Springs for Torsion — Spiral Springs — Conical Springs — Leaf 
Springs — Semi-elliptic Springs — Materials for Springs — References. 

CHAPTER VII 

Belting and Pulleys 147 

Leather Belting — Rubber Belting — Textile Belting — Steel Belting 
— Belt Fastenings — Tensions in Belts — Relation between Tight and 
Loose Tensions — Coefficient of Friction — Maximum Allowable 
Tension — Selection of Belt Size — Taylor's Experiments on Belting 
— Tandem-belt Transmission — Tension Pulleys — Types of Pulleys 
— Transmitting Capacity of Pulleys — Proportions of Pulleys — 
Tight and Loose Pulleys — Types of V Belts — Force Analysis of V 
Belting — References . 

CHAPTER VIII 

Manila Rope Transmission 17 

Manila Hoisting Rope — Sheave Diameters — Stresses in Hoisting 
Rope — Analysis of Hoisting Tackle — Experimental Data on 
Hoisting Tackle — Multiple System — Continuous System — Manila 
Transmission Rope — Sheaves — Relation between Tight and Loose 
Tensions — Force Analysis of a Manila Rope Transmissions-Sheave 
Pressures — Sag of Rope — Efficiency of Manila Rope Drives — 
Selection of Rope — Cotton Rope Transmission — References. 

CHAPTER IX 

Wire Rope Transmission '. 195 

Relation between Load and Effort — Stresses Due to Starting and 
Stopping — Stresses Due to Bending — Stresses Due to Slack — 
Selection of Rope— Hoisting Tackle — Hoisting Sheaves and Drums 
— Design of Crane Drums — Conical Drums — Flat Wire Ropes — 
Single Loop System — Wire Transmission Rope — Transmission 
Sheaves— Stresses in Wire Rope — Sag of Wire Rope — References. 



CONTENTS IX 

CHAPTER X 

Page 

Chains and Sprockets . 216 

Coil Chain — Stud-link Chain — Chain Drums and Anchors — Chain 
Sheaves — Relation between P and Q — Analysis of a Chain Block 
— Detachable Chain — Strength of Detachable Chain — Closed-joint 
Chain — Strength of Closed-joint Chain — Sprockets for Detachable 
Chain — Relation between Driving and Driven Sprockets— Tooth 
Form — Rim, Tooth, and Arm Proportions — Block Chains — 
Sprockets for Block Chains — Selection of Block Chains — Roller 
Chains — Sprockets for Roller Chains — Length of Roller Chain — 
Silent Chains — Coventry Chain — Whitney Chain — Link Belt 
Chain — Morse Chain — Strength of Silent Chain — Sprockets for 
Silent Chain — Spring-cushioned Sprockets — References. 

CHAPTER XI 

Friction Gearing 259 

Experimental Results — Plain Spur Frictions — Applications of Spur 
Frictions— Analysis of a Drop Hammer— Grooved Spur Frictions 
— Starting Conditions of Bevel Frictions — Running Conditions — 
Force Analysis of Crown Frictions — Bearing Pressures Due to 
Crown Frictions — Friction Spindle Press — Curve Described by the 
Flywheel — Pressure Developed by a Friction Spindle Press — 
Double Crown Frictions — Efficiency of Crown Friction Gearing — 
Thrust Bearing for Friction Gearing — Starting Loads — References. 

CHAPTER XII 

Spur Gearing 280 

Definitions — Tooth Curves — Methods of Manufacture — Involute 
System — Laying Out the Involute Tooth — Standard Involute 
Cutters — Action of Involute Teeth — Cycloidal System — Form of 
the Cycloidal Tooth — Laying Out the Cycloidal Tooth — Standard 
Cycloidal Cutters — Action of Cycloidal Teeth— Strength of Cast 
Teeth — Strength of Cut Teeth — Materials and Safe Working 
Stresses — Rawhide Gears — Fabroil Gears — Bakelite Micarta-D 
Gears — Large Gears — Gear Wheel Proportions — Methods of 
Strengthening Gear Teeth — Special Gears — Efficiency of Spur 
Gearing — References . 

CHAPTER XIII 

Bevel Gearing 322 

Methods of Manufacture — Form of Teeth — Definitions — Acute- 
angle Bevel Gears — Obtuse-angle Bevel Gears — Right-angle Bevel 
Gears — General Assumptions Regarding the Strength of Bevel 
Gears— Strength of Cast Teeth— Strength of Cut Teeth— Method 
of Procedure in Problems — Resultant Tooth Pressure — Bearing 



x CONTENTS 

Page 
Pressures and Thrusts — Gear Wheel Proportions — Non-metallic 
Bevel Gears — Mounting Bevel Gears — Spiral Bevel Gears — 
Advantages and Disadvantages of Spiral Bevel Gears — Bearing 
Loads and Thrusts Due to Spiral Bevel Gears — Experimental 
Results — Skew Bevel Gears — References. 

CHAPTER XIV 

Screw Gearing 350 

Types of Helical Gears — Advantages of Double-helical Gears — 
Applications of Double-helical Gears — Tooth Systems — Strength of 
Double-helical Teeth — Materials for Helical Gearing — Double-heli- 
cal Gear Construction — Mounting of Double-helical Gears — 
Circular Herringbone Gears — Straight Worm Gearing — Hind- 
ley Worm Gearing — Materials for Worm Gearing — Tooth Forms 
— Load Capacity — Strength of Worm Gear Teeth — Force Analysis 
of Worm Gearing — Bearing Pressures — Worm and Gear Construc- 
tion — Sellers Worm and Rack — Worm Gear Mounting .—Tandem 
Worm Gears — Experimental Results on Worm Gearing — 
References. 

CHAPTER XV 

Couplings 383 

Flange Coupling — Marine Type of Flange Coupling — Compression 
Coupling — Roller Coupling — Oldham's Coupling — Universal Joint 
. — Leather-link Coupling — Leather-laced Coupling — Francke Coup- 
ling — Nuttall Coupling — Clark Coupling — Kerr Coupling — Rolling 
Mill Coupling — Positive Clutch— Analysis of Jaw Clutches — 
References. 

CHAPTER XVI 

Friction Clutches 405 

Requirements of a Friction Clutch — Materials for Contact Sur- 
faces — Classification of Friction Clutches — Single-cone Clutch — 
Double-cone Clutch — Force Analysis of a Single-cone Clutch — A 
Study of Cone Clutches — Experimental Investigations of a Cone 
Clutch — Analysis of a Double-cone Clutch — Smoothness of 
Engagement of Cone Clutches — Clutch Brakes — Single-disc 
Clutch — Hydraulically Operated Disc Clutch — Slip Coupling — 
Multiple-disc Clutches — Force Analysis of a Disc Clutch — A Study 
of Disc Clutches — Hele-Shaw Clutch — Ideal Multi-cone Clutch — 
Moore and White Clutch — Transmission Block Clutches — Analysis 
of Block Clutches — Machine-tool Split-ring Clutches— Analysis of 
a Split-ring Clutch — Study of Split-ring Clutches — -Types of Band 
Clutches — Analysis of a Band Clutch — Horton Clutch — Require- 
ments of an Engaging Mechanism — Analysis of Engaging Mechan- 
isms — References . 



CONTENTS xi 

CHAPTER XVII 

Pagb 

Brakes- . 462 

General Equations — Classification — Single- and Double-block 
Brakes — Analysis of Block Brakes — Graphical Analysis of a 
Double-block Brake — Simple Band Brakes — Band Brakes for 
Rotation in Both. Directions — Differential Band Brakes — Conical 
Brakes — Disc Brakes — Worm-gear Hoist Brakes — Crane Disc 
Brakes — Crane Coil Brakes — Cam Brake — Force Analysis of an 
Automatic Brake — Disposal of Heat — References. 

CHAPTER XVIII 

Shafting 489 

Materials — Method of Manufacture — Commercial Sizes of Shaft- 
ing — Simple Bending — Simple Twisting — Combined Twisting and 
Bending — Method of Application — Combined Twisting and Com- 
pression — Bending Moments — Crane Drum Shaft — Shaft Support- 
ing Two Normal Loads between the Bearings— Shaft Supporting 
Two Normal Loads with One Bearing between the Loads — Shaft 
Supporting One Normal and One Inclined Load between the Bear- 
ings — Two-bearing Shafts Supporting Three Loads — Hollow Shafts 
— Effect of Key-seats upon the Strength of Shafts — Effect of Key- 
seats upon the Stiffness of Shafts — References. 

CHAPTER XIX 

Jour nals, Bearings, and Lubrication 513 

Types of Bearings — General Considerations — Selection of Bearing 
Materials — Provisions for Lubrication — Adjustments for Wear — 
Adjustments for Alignment — Bearing Pressures — Relation between 
Length and Diameter — Radiating Capacity of Bearings — Coeffi- 
cient of Friction — Design Formulas — Temperature of Bearings — 
Strength and Stiffness of Journals — Design of Bearing Caps and 
Bolts — Work Lost Due to the Friction on a Cylindrical Journal — 
Work Lost Due to the Friction on a Conical Journal — Proportions 
of Journal Bearings — Solid Bearing with Thrust Washers — Collar 
Thrust Bearings — Step Bearings — Work Lost Due to Pivot Fric- 
tion — Work Lost in a Collar Thrust Bearing — Analysis of a Flat 
Pivot — Tower's Experiments on Thrust Bearings — Schiele Pivot 
— References. 

CHAPTER XX 

Bearings with Rolling Contact 556 

Requirements of Rolling Contact — Classification — Radial Bearings 
having Cylindrical Rollers — Radial Bearings having Conical 
Rollers — Radial Bearings having Flexible Rollers — Thrust Bearing 
having Cylindrical Rollers — Thrust Bearing having Conical Rollers 



CONTENTS 

— Allowable Bearing Pressures and Coefficients of Friction — 
Roller Bearing Data — Mounting of Roller Bearings — Forms of 
Ball Bearing Raceways — Experimental Conclusions of Stribeck 
— Radial Ball Bearings — Thrust Ball Bearings — Combined Radial 
and Thrust Bearing — Allowable Bearing Pressures — Coefficient 
of Friction — Ball Bearing Data — Mounting Ball Bearings — 
References. 



TABLES 

Table • Page 

1. Poisson's Ratio 6 

2. Modulii of Resilience for Steel in Tension 8 

3. Values of Constant a 21 

4. Values of Coefficient of Linear Expansion 22 

5. Suggested Factors of Safety 23 

6. Average Physical Properties of Principal Materials 24 

7. Specifications of Pig Iron 28 

8. General Specifications of Pig Iron 29 

9. Efficiency of Boiler Joints 57 

10. Ultimate Shearing Stresses in Rivets 58 

11. Thickness of Shell and Dome Plates after Flanging 58 

12. Thickness of Butt Joint Cover Plates 58 

13. Recommended Sizes of Rivet Holes 59 

14. Tension Members 71 

15. United States Standard Bolts and Nuts 78 

16. Proportions of Sellers Square Threads 79 

17. Proportions of Acme Standard Threads 80 

18. Coupling Bolts 81 

19. S. A. E. Standard Bolts and Nuts 82 

20. Standard Cap Screws 84 

21. Standard Machine Screws 86 

22. Safe Holding Capacities of Set Screws 87 

23. Plain Lock Washers , 94 

24. Coefficients of Friction for Square Threaded Screws 107 

25. Bearing Pressures on Power Screws 108 

26. Dimensions of Woodruff Keys 112 

27. Diameters of Shafts and Suitable Woodruff Keys . . . . , . .113 

28. Round Keys and Taper Pins 115 

29. Dimensions of Gib-head Keys 119 

30. S. A. E. Drop Forged Rod and Yoke Ends 127 

31. B. & S. Drop Forged Rod and Yoke Ends 128 

32. Values of Coefficients K, K x , K 2 , and K 3 . . . 133 

33. Values of Coefficients K 4 ,K 5 ,K 6 , and K 7 . 134 

34. Values of Coefficients K 8 and K 9 136 

35. Results of Test on Leather Belting 149 

36. Strength of Leather Belt Joints 155 

37. Average Ultimate Strength of Leather Belting 160 

38. Working Stresses for Leather Belting 160 

39. Comparative Transmitting Capacities of Pulleys 166 

40. Proportions of Extra-heavy Cast-iron Pulleys 167 

41. Manila Rope 176 

42. Hoisting Tackle Reefed with Manila Rope 179 

43. Dimensions of Grooves for Manila Rope Sheaves 184 

xiii 



xiv TABLES 

Table Page 

44. Dimensions of Grooves for Manila Rope Sheaves 184 

45. Tensions due to Slack as Shown by Dynamometer 200 

46. SteelWireRope 203 

47. Hoisting Tackle Reefed with Wire Rope 205 

48. General Dimensions of Wire Rope Sheaves 206 

49. Dimensions of Grooves for Wire Rope Drums 207 

50. Coefficients of Friction for Wire Rope 214 

51. Hoisting Chains 217 

52. Dimensions of Grooves for Chain Drums 218 

53. Dimensions of Plain Chain Sheaves 221 

54. Ewart Detachable Chain 230 

55. Closed-joint Conveyor and Power Chains 232 

.56. Union Steel Chains 233 

57. Sprocket Teeth Factors 236 

58. Diamond Block Chain 240 

59. Diamond Roller Chains 244 

60. Design Data for Morse Silent Chain Drives 252 

61. Whitney Silent Chains 253 

62. Horse Power Transmitted by Link Belt Silent Chain 254 

63. General Proportions of Link Belt Sprockets 256 

64'. Experimental Data Pertaining to Friction Gearing 260 

65. Radii for 15° Involute Teeth 286 

66. Brown and Sharpe Standard Involute Cutters 287 

67. Radii for Cycloidal Teeth . 290 

68. Brown and Sharpe Standard Cycloidal Cutters ........ 291 

69. Lewis Factors for Gearing ; . . 297 

70. Lewis Factors for Stub Teeth 298 

71. Proportions of Cut Teeth 299 

72. Values of S for Various Materials 302 

73. Data Pertaining to Rawhide Gears 304 

74. Dimensions of Gear Hubs 310 

75. Strength of Gear Teeth used by C. W. Hunt 312 

76. Dimensions of the Fellows Stub Teeth 313 

77. Constants for Gleason Unequal Addendum Teeth 314 

78. Experimental Data Pertaining to Bevel Gears 349 

79. Proportions of Helical Teeth 354 

80. Proportions of Fawcus Double-helical Teeth 354 

81. Values of Coefficient K as recommended by W. C. Bates .... 356 

82. Cramp's Gear Bronzes 367 

83. Standard 29° Worm Threads 368 

84. Results of Tests on Cast-iron Worm Gearing 381 

85. Proportions of Westinghouse Flange Couplings 387 

86. Dimensions of Clamp Couplings 388 

87. Dimensions of Bocorselski's UniversalJoints 391 

88. Dimensions of Merchant & Evans Universal Joints 392 

89. Data Pertaining to Leather Link Couplings 394 

90. Data Pertaining to Leather Laced Couplings 396 

91i Data Pertaining to Francke Couplings 398 



TABLES xv 

Table Page 

92. Service Factors for Francke Couplings 398 

93. Proportions of Slip Couplings '. . 431 

94. Data Pertaining to Various Types of Disc Clutches 437 

95. Fiber Stresses at the Elastic Limit 499 

96. Allowable Bearing Pressures 528 

97. Relation between Length and Diameter of Bearings 529 

98. Dimensions of Rigid Post Bearings 541 

99. Coefficients of Friction for Collar Thrust Bearings 552 

100. Coefficients of Friction for Step Bearings 552 

101. Data Pertaining to Norma Roller Bearings 563 

102. Data Pertaining to Hyatt High Duty Bearings 564 

103. Crushing Strength of Tool Steel Balls 575 

104. Data Pertaining to Hess-Bright Radial Ball Bearings 577 

105. Data Pertaining to S. K. F. Radial Ball Bearings 579 

106. Data Pertaining to F. & S. Thrust Ball Bearings 581 

107. Data Pertaining to Gurney Radio-thrust Ball Bearings ... . 585 



MACHINE DESIGN 

CHAPTER I 
STRESSES AND STRAINS IN MACHINE PARTS 

1. Forces. — The object of a machine is to transmit motion 
through its various links to some particular part where useful 
work is to be done. The transmission of this motion gives rise 
to forces which must be resisted by the parts of the machine 
through which the forces are acting. 

The forces acting upon the various machine parts may be 
classified as follows: 

(a) Useful forces. — In doing the useful work for which the 
machine is intended, the various parts are subjected to certain 
forces; for example, the parts of a shaper must transmit the forces 
produced by the resistance offered to the tool by the material to 
be cut. 

(b) Dead weight forces.— Dead weight forces are those due to 
the weights of the individual parts in a machine. Generally 
these forces are not considered in the design of a machine, except 
in cranes and machines having large gears and flywheels. In the 
design of roof trusses, bridges, structural steel towers, floors, etc., 
the dead weights are always considered, because they form a con- 
siderable part of the total load coming upon the members. 

(c) Frictiondl forces.- — -Forces called forth by the frictional 
resistances between the machine parts are designated as fric- 
tional forces. In certain classes of machinery such as hoists 
employing screws, a large amount of work is consumed by fric- 
tion; hence the various machine elements must transmit this 
energy in addition to energy required for useful work. 

(d) Forces due to change of velocity. — ^Frequently the motion of 
machine parts changes in direction, thus causing forces that must 
be considered ; for example, the rim of a rapidly rotating flywheel 
or pulley is subjected to rather heavy centrifugal forces. Another 
example is given by the whipping action of the connecting rod of 
a high-speed engine; the stresses arising from the reversal of 



2 PRINCIPLES GOVERNING DESIGN [Chap. I 

direction may be far in excess of those due to the steam pres- 
sure on the piston. In general, whenever the velocity of a 
machine part changes rapidly heavy stresses are set up due to 
the accelerating and retarding action. Forces due to a change 
of velocity are frequently called inertia forces. 

(e) Forces due to expansion and contraction. — In certain struc- 
tures, as boilers, the forces due to the expansion and contraction 
caused by heat must be considered. Not infrequently heavy 
bending stresses are induced in members by expansion and 
contraction. 

(/) In addition to the various forces discussed in the preceding 
paragraphs others exist, such as the following: (1) forces due to 
the reduction of area caused by the deterioration of the material; 
(2) force due to the use of non-homogeneous material; (3) forces 
due to poor workmanship. 

Some of the above-mentioned conditions need not be considered 
at all, but it is well that they all be kept in mind when undertak- 
ing the design of a new machine. It is evident from this 
brief discussion, that before the designer can select a suitable 
material or determine the proportions of the various elements, he 
must make a careful analysis of the external forces and their 
effects. 

2. Principles Governing Design. — The design of machine parts 
may be approached by either of the following methods : 

(a) Strength alone is the basis of design ; that is, the parts are 
made strong enough to resist the stresses developed in them, and 
as long as no rupture occurs the machine parts designed in this 
way fulfill their purpose. As an illustration, the design of a gear 
transmitting a given horse power at a certain speed is in general 
satisfactory so long as the various component parts of the gear, 
such as the teeth, rim and arms, do not rupture. 

(b) Stiffness in addition to strength is taken into consideration ; 
that is, machine parts must be made rigid enough to perform their 
function without too much distortion. Stiffness is essential in the 
design of all the important elements of a machine tool, as without 
rigidity the machine is not capable of producing work having the 
desired degree of accuracy. A grinding machine is a good illus- 
tration in the design of which the consideration of stiffness is 
paramount. 

The criterion discussed in (b) is important, and whenever pos- 
sible a study of the deflections of the various members of the ma- 



Art. 3] STRESS-STRAIN DIAGRAM 3 

chine should be attempted. This study, in the majority of cases, 
is very difficult to carry out, as the deflections are not readily 
calculated; as a matter of fact, in many cases it is impossible to 
calculate such deflections with our present knowledge of the sub- 
ject of "Strength of Materials." Frequently the determina- 
tion of the stresses induced in certain machine parts is beyond 
calculation and in such cases as well as those mentioned above, 
experience together with precedent must be relied upon to suggest 
the proper proportions to be used. 

STRESS, STRAIN, AND ELASTICITY 

3. Stress and Strain. — The external forces or loads coming 
upon the members of a machine cause the latter to undergo a 
deformation or change of form, the amount of which is called a 
strain. Now within the member that is thus deformed, a certain 
internal force is produced in the material which will resist this 
strain. This internal force is called a stress, and may be defined 
as the internal resistance which the particles of the material offer 
to the external force. A designer should evidently have a knowl- 
edge of the stresses and strains induced in a material subjected to 
an external force, and without such knowledge it is impossible for 
him to produce a well-designed machine. Information pertain- 
ing to stresses and strains is derived from tests of materials. The 
following articles give briefly some of the results of such tests. 

4. Stress-strain Diagram. — The relation existing between the 
unit stresses and unit strains for any particular material is shown 
best by means of a diagram. This diagram is based upon the ob- 
servations and calculations derived from experiments, and is 
constructed by plotting upon rectangular coordinates the unit 
strains against the unit stresses, the latter as ordinates and the 
former as abscissae. In Fig. 1 is shown such a diagram. The 
plot shown represents the results of a tensile test on a soft grade of 
machinery steel. The results of a compression test on any mate- 
rial may be plotted in a similar manner. 

An inspection of the plot in Fig. 1 shows that up to a certain 
point B the stress-strain diagram is practically a straight line; 
that is, unit stress is proportional to unit strain. The law just 
stated is known as Hooke's Law. The stress corresponding to 
the point B is known by the term, "limit of proportionality" or 
better still "proportional elastic limit. " At the point C there is a 



STRESS-STRAIN DIAGRAM 



[Chap. I 



well-defined break in the diagram, thus showing a sudden and con- 
siderable increase of strain without an appreciable increase of 
stress; in other words, this point indicates a change in the condi- 
tion of the material, namely from one of almost perfect elasticity 
to one of considerable plasticity. The point C is called the yield 
point, and is found only on the stress-strain diagrams of the duc- 
tile materials. In testing ductile materials the stress correspond- 
ing to the yield point is obtained by observing the load on the 
scale beam of the machine at the instant the beam takes a sudden 
drop. 



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Unit Strain 
Fig. 1. 



Another term used considerably and frequently applied to the 
stress corresponding to the point B in Fig. 1 is the elastic limit. 
Various definitions have been proposed for this term, and the 
following is considered about the best: By the elastic limit is 
meant the unit stress below which the deformation or strain dis- 
appears completely upon removal of the stress; in other words, no 
permanent set can be detected. The determination of the elastic 
limit experimentally requires instruments of high precision, and 
due to the repeated application and release of the stress that is 
necessary, such tests require a great amount of time. In general 
it is assumed that there is but little difference between the elastic 
limit and the stress corresponding to the limit of proportionality; 
and since the latter can be determined more readily, it may be 



Akt. 5] 



MODULUS OF ELASTICITY 



used by designers as a means of getting at the probable elastic 
limit of a material. 

Referring again to Fig. 1, it is evident that as the stress in- 
creases, the deformation increases, until finally rupture of the 
test piece occurs. The external load required to break the test 
piece divided by the original area of cross-section of the bar is 
called the ultimate strength. 

5. Modulus of Elasticity. — In order to determine the strain for 
any known load acting upon a given material, it is convenient to 
make use of the so-called modulus of elasticity. This is defined as 



2 60 



40 



z 



2Z 



Spring Steel 

0. 65 C Steel 

Cold Rolled 
Medium Steel 

Soft Steel 



0.001 0.002 0.003 

Unit Strain 
Fig. 2. 



the ratio of unit stress to unit strain, a value of which is readily 
obtained from that part of the stress-strain diagram below the 
point B; in other words, the slope of the line AB gives the value 
of the modulus of elasticity. Representing this modulus of elas- 
ticity for tension by the symbol E t , the statement just made may 
be expressed algebraically by the following equation : 

E, = f ■ (1) 

in which S t denotes the unit stress and 8 the unit deformation; 
hence E t is some quantity expressed in the same units as S f , 
namely in pounds per square inch. 



POISSON'S RATIO 



[Chap. I 



In Fig. 2 are shown stress-strain diagrams for several grades 
of steel, which seem to indicate that the modulus of elasticity is 
practically the same for all grades of steel. According to the 
results obtained by various authorities the numerical value of 
the modulus of elasticity for steel varies from 28,000,000 to 32,- 
000,000 pounds per square inch. The modulus of elasticity is also 
a measure of the stiffness or rigidity of a material, and from Fig. 2 
it is evident that a machine part made of soft steel will be just 
as rigid as if it were made of an alloy steel, provided the stresses 
in the member due to the external load are kept below the 
limit of proportionality. However, the part when made of high- 
carbon steel will be much stronger than that made of soft steel. 
It has been suggested by certain machine-tool builders that ex- 
cessive deflections of spindles and shafts may be reduced by the 
use of an alloy steel in place of a 25-point carbon open-hearth 
steel, but upon actual trial it was found that the trouble was not 
remedied. The failure of the alloy steel to decrease the deflec- 
tion, is due to the fact that the modulus of elasticity and not the 
strength of the steel is the measure of its rigidity. 

6. Poisson's Ratio. — When a bar is extended or compressed the 
transverse dimension as well as the length are changed slightly. 

Experimental data show that 
the ratio of the transverse unit 
strain to the unit change in 
length is practically constant. 
This ratio is called Poisson's 
ratio, average values of which, 
collected from various sources, 
are given in Table 1. 

7. Resilience. — Referring 
to Fig. 1, it is evident that the 
area under the complete curve represents the work done in ruptur- 
ing the test specimen, while that under the diagram up to any 
assumed point on the curve represents the work done in stretch- 
ing the specimen an amount equal to the deformation correspond- 
ing to the assumed point. If this assumed point be taken so 
that the stress corresponding to it is equal to the elastic limit, 
then the area under that part of the diagram represents the work 
done in producing a strain corresponding to that at the elastic 
limit. The energy thus spent is called .resilience, and is repre- 



Table 1. — Poisson's 


Ratio 


Material 


Poisson's 
ratio 




0.270 




0.278 


cu i 1 Hard 
Steel {Mild 


0.295 
0.303 
0.340 




0.350 







Art. 7] RESILIENCE 7 

sented in Fig. 1 by the triangular area AED. Since the area of 
this triangle is }i(AE X ED), it follows that 

Resilience = -y- X -^- = ^ i (2) 

in which A denotes the cross-sectional area of the test specimen, 

L its length, and S e the stress at the elastic limit. 

If the specimen has a cross-sectional area of one square inch 

and a length of one inch, then the second member of (2) reduces 

S 2 
to ^-j^- This magnitude is then the unit of resilience and is 

z hit 

called the modulus of resilience, a quantity which is useful for 
comparing the capacity various materials have for resisting shock. 
As mentioned in Art. 5, the modulus of elasticity in tension is 
practically constant for the various kinds of carbon and alloy 
steels; hence it follows from (2), that the modulii of resilience 
of two steels are to each other as the squares of the stresses at 
their elastic limits. From this fact it is apparent that the higher 
carbon steels have a greater capacity for resisting shock than 
those of lower carbon content, since their elastic limits are higher 
as shown in Fig. 2. 

Unfortunately writers on "Strength of Materials" have paid 
but little attention to the actual values of the modulus of resilience, 
and consequently information pertaining thereto is not plentiful. 
The values given in Table 2 were calculated by means of the 
expression for the modulus, and may serve as a guide in the proper 
selection of a shock-resisting material. The stresses at the 
elastic limit given in Table 2 were collected from various sources, 
and in all probability in the majority of cases, the yield point 
instead of the elastic limit is referred to. The error introduced 
by substituting the stress at the yield point for that at the 
elastic limit is not of great consequence since the values given 
in Table 2 serve merely as a guide. 

It should be noted that the preceding discussion of resilience 
applies to the stress-strain diagram given in Fig. 1 which repre- 
sents the result of a tensile test; however, the formula for the 
modulus of resilience applies also to direct compressive or shear- 
ing stresses, provided the modulus of elasticity and the stress at 
the elastic limit are given their appropriate values. 



TENSILE STRESS [Chap. I 

Table 2. — Moduli of Resilience for Steel in Tension 



Type of steel 



Elastic 
limit 



Modulus of 
resilience 



Open-hearth 
carbon steels 



Alloy steels 



0.08 per cent. C 
0.15 per cent. C 
0.30 per cent. C 
0.40 per cent. C 
0.50 per cent. C 
0.60 per cent. C 
0.70 per cent. C 
0.80 per cent. C 



25,000 
30,000 
35,000 
41,000 
47,500 
63,500 
70,500 
75,000 



Nickel steel, 2.85 per 
cent. Ni. 



Annealed 



52,000 



Oil-tempered 



121,000 



Chrome steel, oil-tempered. 



127,500 



Carbon vanadium, oil-tempered. 



136,000 



Nickel vanadium, oil-tempered. 



126,250 



Chrome vanadium. 



Annealed 



63,700 



Oil-tempered 



170,000 



10.4 
15.0 
20.4 
28.0 
37.6 
67.2 
82.8 
93.8 



45.1 



244.0 



271.0 



308.0 



266.0 



67.5 



482.0 



SIMPLE STRESSES 

The external forces acting upon a machine part induce various 
kinds of stresses in the material, depending upon the nature of 
these forces. The different kinds of stresses with which a designer 
of machines comes into contact will now be discussed briefly. 

8. Tensile Stress. — A machine member is subjected to a ten- 
sile stress when the external forces acting upon it tend to pull it 
apart. Using the notation given below, the relations existing be- 
tween stress, strain and the external forces for the case of simple 
tension are derived as follows: 

Let A = cross-sectional area of the member. 
E t = modulus of elasticity. 
L = length of the member. 
P = the external force. 
St = unit tensile stress. 
A = total elongation. 



Art. 9] SHEARING STRESS 9 

The area of cross-section of the member multiplied by the 
unit stress gives the total stress induced in the section, and since 
the total stress induced is that due to the pull of the force P, 
it follows that 

S t = j (3) 

From the definition of the modulus of elasticity given in Art. 
5, or from (1), we get 

E t = ^ (4) 

from which the following expression for the total elongation is 
obtained : 

*=H <» 

By means of (5) , it is possible to determine the probable elonga- 
tion of a given member subjected to a load P. This is a very 
desirable thing to do for all tension members of considerable 
length, as very frequently such elongation is limited by the class 
of service for which the proposed machine is intended. 

9. Compressive Stress. — A compressive stress is induced in a 
member when the external forces tend to force the particles of the 
material together. For a short member, in which no buckling 
action is set up by the external forces, the various relations de- 
duced in Art. 8 apply also in this case, provided the appropriate 
values are substituted for the various symbols. If, however, the 
length of the member exceeds say six times the least diameter, 
the stresses induced must be determined by the column formulas, 
which are discussed in Art. 15. 

A kind of compressive stress met with extensively in designing 
machinery is that caused by two surfaces bearing against each 
other; for example, the edges of plates against rivets or pins, or 
keys against the sides of the key-way or key-seat. This kind of 
a stress is usually spoken of as a bearing stress. 

10. Shearing Stress. — A shearing stress is one that is produced 
by the action of external forces whose lines of action are parallel 
and in opposite direction to each other. The relation existing 
between the external force P, area of cross-section A, and the 
shearing stress S s , is similar to (3), or 

Ss = t (6) 



10 SHEARING STRESS [Chap. 1 

If a machine member is twisted by a couple, the stress induced 
in that member is a pure shear, or as it is commonly called, a 
torsional stress. The following discussion establishes the relations 
existing between stress, strain and the external forces for a 
member having a circular cross-section. 

Equating the external moment T to the internal resisting 
moment, we obtain 

T = ^H (7) 

a 

in which J represents the polar moment of inertia and d the di- 
ameter of the member. For any given section the value of J 
may be obtained by means of the relation: 

J = A + /,, (8) 

in which i\ and I 2 represent the rectangular moments of inertia 

of the section about any two axes at right angles to each other, 

through the center of gravity. For a circular cross-section 

7rd 4 
J = 2 7i = ^y, hence (7) becomes 

Trd*S s 



16 



(9) 



The relation between the twisting moment T and the angular 
deflection d of a circular member having a length L is derived in 
the following manner: 

_ 360 SJj , m , 

Es ~ ^JT (10) 

Substituting in (9) the value of S s from (10), we obtain 

6d 4 E s 



584 L 



(11) 



The expression given by (9) is to be used when the member 
must be designed for strength, while (11) is used to proportion 
the member for stiffness. 

11. Stresses Due to Flexure.- — Machine members may be 
subjected to transverse forces which produce stresses of several 
kinds. Such members must be designed by considering the 
effect produced by the combination of these several stresses. A 
simple illustration of a member in which several kinds of stresses 
are induced is an ordinary beam supported at its ends and carry- 
ing a load W at a distance x from the left-hand support. Due 



Art. 12] STRESSES DUE TO FLEXURE 11 

to the load W, the beam will bend downward producing a com- 
pressive stress on the upper or concave side, a tensile stress on the 
lower or convex side, and a shearing stress at right angles to the 
tensile and compressive stresses just mentioned. In calculations 
pertaining to beams, the magnitude of the shearing stress is 
generally small relative to the tensile and compressive stresses, 
and may then be neglected altogether; however, cases may arise 
when the shearing stress must be considered. 

The relation existing between the bending moment produced 
in the beam by the load W, the stress S and the dimensions of the 
cross-section of the beam, is obtained by equating the external 
moment to the internal stress moment; thus 

M = —, (12) 

c 

in which / represents the moment of inertia of the beams cross- 
section, and c the distance from the center of gravity of the sec- 
tion to the outermost fiber. This formula is applicable for de- 
termining the strength of the beam, provided $ is kept within 
the elastic limit. 

Whenever a beam is to be designed for stiffness the following 
general formula may be used : 

u = m% (13) 

The expression given by (13) is the fundamental equation by 
means of which the deflection of any beam may be obtained. 
The method of procedure is to determine, for the case considered, 
an expression for the bending moment M in terms of x, and after 
substituting it in (13), integrate twice and solve for the vertical 
deflection y of the beam.. 



COMBINED STRESSES 

12. Flexure Combined with Direct Stress. — In structures such 
as bridges and roofs, the members are, in general, pieces that are 
acted upon by equal and opposite forces. There being no motion 
at the joints, it is properly assumed that such members are cen- 
trally loaded, thus producing a uniformly distributed stress in 
the material. When, however, we deal with machine parts, 
central loading is the exception rather than the rule. Even in 
the ordinary connecting link used merely to transmit motion, 



12 



STRAIGHT PRISMATIC BAR 



[Chap. I 



the friction between the pin and its bearing in the link causes a 
shifting of the line of action by an appreciable amount, thus 
subjecting the link to a flexural stress in addition to the direct 
stress. 

In order to determine the distribution of stress in any right 
section of a member subjected to flexure combined with direct 
stress, and thence to find the maximum intensity of stress, the 
following analysis and discussion is recommended. Attention is 
called to the fact that the expressions given are strictly applicable 
only to the following types of members : 

(a) Short as well as long tension members that are straight. 

(6) Short and straight compression members. 

13. Straight Prismatic Bar. — In Fig. 3 is shown a straight pris- 
matic bar so loaded that the line of action of the external force 
P is parallel to the axis AB and at a dis- 
tance e from it. We are to determine the 
distribution of stress in any right section 
as CD and thence to find the maximum in- 
tensity of stress. Consider the portion of 
the bar above CD as a free body, and at 
the center of the section insert two op- 
posite forces OM and ON acting along 
the axis AB, and equal to the external 
force P. These forces being equal and 
opposite, do not affect the equilibrium 
of the system. We have thus replaced 
the single external force by the central 
force OM and a couple consisting of the 
equal and opposite forces P and ON. The 
arm of the couple is e and its moment is 
Pe. The single force OM must be 
balanced by a stress in the section CD; 
and since OM has the axis AB as its line of action, this stress 
is uniformly distributed over the cross-section. Denoting the 
intensity of this stress by S' t , and the area of the section by A, 
we have 

P 



A 


P 


\ 


-e- 
M 


JM 


4 

N ^ 


B 


P 



Fig. 3. 



s' t 



(14) 



If the intensity S t , be denoted in Fig. 3 by CE, the line EF 
parallel to CD will indicate graphically the uniform distribution 
of stress over the section. 



Art. 13] STRAIGHT PRISMATIC BAR 13 

The couple of moment Pe tends to give the body under con- 
sideration a counter-clockwise rotation. Evidently this couple 
must be balanced by a stress with an equal moment and of 
opposite sense. The fibers to the right of A B will be subjected 
to tensile stress and those to the left to compressive stress. De- 
noting the intensity of the flexural stress at D by S", and the 

section modulus of the section by — , then 

Ct 

n Pec, 
S'/ = ^ (15) 

Denoting the intensity of the flexural stress at the point C 

by S", and the section modulus of the section by — , we get 

// Pec 
S" = ~ c (16) 

The law of distribution of the stress induced by the couple Pe 
is represented graphically by the line GH. Evidently the maxi- 
mum intensity of tensile stress occurs at the point D, and its 
magnitude is obtained by adding (14) and (15), or 

The maximum compression stress occurs at the point C, and 
its magnitude is given by the following expression: 

fl.-zPr'-i] (18) 

Equations (17) and (18) are not strictly exact, since the flexural 
stresses S t and S c do not represent actual direct stresses and 
therefore should not be combined directly with the true direct 
stress S' t . The difference between these stresses may be con- 
siderable for materials in which the rates of deformation due to 
tension and compression are not equal, as in cast iron, brass, and 
wood. A better method would be to express S" and S" in 
terms of S' t before combining them with the latter. In general, 
to express a stress due to flexure in terms of a direct stress, multi- 
ply the former by the ratio that the direct stress of the given 
material bears to the transverse stress. 

In the analysis just given the external force P produces a direct 
tensile stress over the area A; however, the various formulas 
derived above apply to the condition when the force P is reversed, 



14 



OFFSET CONNECTING LINK 



[Chap. I 



namely producing a direct compressive stress, providing the 
proper symbols are used. 

14. Offset Connecting Link. — A case of frequent occurrence 
in the design of machine parts is the offset connecting link shown 
in Fig. 4. The circumstances are such that it is not practicable 
to make the link straight, and the axis of a cross-section, as CD, 
lies at a distance e from the line AB, 
which joins the centers of the pins and is, 
therefore, neglecting friction, the line of 
action of the external forces. Let bo and 
h denote the dimensions of the rectan- 
gular cross-section of the link if straight 
and centrally loaded; and let b and h de- 
note the corresponding dimensions of the 
eccentrically loaded section at CD. 

For the straight link the intensity of 
the uniformly distributed stress is 

s » = w (19) 

For the offset link the maximum intensity 
of stress in the section CD as calculated 
by means of (17) is 

If we impose the condition that S t shall not exceed So, we have 




bh > 6« 



„4f + i] 



(21) 



Let mho denote the distance of the right-hand edge of the cross- 
section CD from AB, the line of action of the external forces; 
this is to be taken positive when measured from AB to the right, 
that is, when AB cuts the section in question, and negative when 
measured from AS to the left. Then the eccentricity is 



mh< 



(22) 



Substituting this value in (21), we have finally 

Qmho 



bh i w , [ 4 - <""*•] 



(23) 



Art. 15] STRESSES IN COLUMNS 15 

A discussion of (23) leads to some interesting results. For 
given values of boh and m, we may vary b and h as we choose, 
subject to the restriction expressed by (23). Economy of 
material is obtained by making the product bh and, therefore, 

the expression 4 r — as small as possible. If m is posi- 
tive, that is, if the section is cut by the line of action of the 
forces, this requirement is met by making h as small as possible; 
on the other hand, if m is negative, that is, if the section CD lies 
wholly outside of the line of action of P, the product bh is made 
a minimum by making h as large as possible. In other words, 
when m is positive, keep the width h as small as possible and 
increase the area of the section by increasing the thickness 6; 
when m is negative, keep the thickness b small and add to the 
area of the section by increasing the width h. This principle 
is of importance in the design of the C-shaped frames of punches, 
shears, presses and riveters. 

When m = 0, that is when the edge of the section coincides with 
the line of action AB, (23) reduces to bh ^ 4 b h . The area of 
section bh must be at least four times the area of section b h 0) 
independent of the relative dimensions of the section. 

15. Stresses in Columns. — As stated in Art. 9, the formulas for 
short compression members are not applicable to centrally loaded 
compression members whose length is more than six times its 
least diameter. Due to the action of the external load, such a 
member will deflect laterally, thus inducing bending stresses 
in addition to the direct stress. 

(a) Ritter's formula. — Many formulas have been proposed 
for determining the permissible working stress in a column of 
given dimensions. Some of these are based upon the results ob- 
tained from tests on actual columns, while others are based on 
theory. In 1873, Ritter proposed a rational formula, by means of 
which the value for the mean intensity of permissible compressive 
stress in a long column could be determined. This formula, 
given by (24) is used generally by designers of machine parts : 

* - i = "Jfe: (24) 

+ n#7r 2 r 2 
in which 

A = area of cross-section. 
E = coefficient of elasticity. 



16 STRESSES IN COLUMNS [Chap. I 

L = the unbraced length of the column in inches. 
P = the external load on the column. 
S c = the greatest compressive stress on the concave side. 
S e = unit stress at the elastic limit. 
n = a constant. 
r = least radius of gyration of the cross-section. 

The strength of a column is affected by the condition of the 
ends, that is the method of supporting and holding the columns. 
In (24) this fact is taken care of by the factor n, which may have 
the following values, taken from Merriman's "Mechanics of 
Materials." 

1. For a column fixed at one end and free at the other, n = 0.25. 

2. For a column having both ends free but guided, n = 1. 

3. For a column having one end fixed and the other guided, 

n = 2.25. 

4. For a column having both ends fixed n = 4. 
(b) Straight line formula.. — A formula used very extensively 

by structural designers is that proposed by Mr. Thos. H. Johnson, 
and is known as the straight line formula. It is not a rational 
formula, but is based on the results of tests. Using the same 
notation as in the preceding article, Johnson's straight line 
formula for the mean intensity of permissible compressive stress is 

Sc = j = Sc-f> (25) 



in which C is a coefficient whose value may be determined by the 
following expression: 

n _ S c 

C ~3 



Vra (26 > 



The factor n in (26) has the same values as those used in con- 
nection with Ritter's formula given above. 

The straight line formula has no advantage over the Ritter 
formula as far as simplicity is concerned, except possibly in a 
series of calculations in which the value of C remains constant, 
as, for example, in designing the compression members of roof 
trusses in which the same material is used throughout. For a 
more complete discussion of the above formulas the reader is 
referred to Mr. Johnson's paper which appeared in the Transac- 
tions of the American Society of Civil Engineers for July, 1886. 



Art. 17] COMBINED STRESSES 17 

16. Eccentric Loading of Columns. — Not • infrequently a de- 
signer is called upon to design a column in which the external 
force P is applied to one side of the gravity axis of the column; in 
other words, the column is loaded eccentrically. A common 
method in use for calculating the stresses in such a column 
consists of adding together the following stresses: 

(a) The stress due to the column action as determined by 

means of the Ritter formula, or 7 H „ „' • 

Pec 
(0) The flexural stress due to the eccentricity, namely -j~i > 

in which c is the distance from the gravity axis of the column to 
the outer fiber on the concave side, and e is the eccentricity of the 
external force P, including the deflection of the column due to the 
load. For working stresses used in designing machine members, 

the deflections of columns having a slenderness ratio - of less 

than 120 are of little consequence and for that reason may be 
neglected, thus simplifying the calculations. 

By adding the two stresses we find that the expression for the 
maximum compressive stress in an eccentrically loaded column is 



S e 



P\~, , SeL 



r\ . sjj* , eel /0 _ x 

L 1 + 53Bi- + ?J (27) 



17. Shearing Combined with Tension or Compression. — Many 
machine members are acted upon by external forces that produce 
a direct tensile or compressive stress in addition to a direct shear- 
ing stress at right angles to the former. The combination of 
these direct stresses produces similar stresses, the magnitudes 
of which may be arrived at by the following expressions taken 
from Merriman's "Mechanics of Materials:" 

Maximum tensile stress 
Maximum compressive stress 

Maximum shearing stress 

These formulas will be found useful in arriving at the resultant 
stresses in machine members subjected to torsion combined with 
bending or direct compression. Such cases will be discussed in 
the chapter .on shafting. 



s, 

2 T 


-^ 


+ f 




(28) 


2 ^ 


-J&" 


+ f 


(29) 


4 s - 


^ 4 


2 or yjs s * 


+ s ± 


(30) 



18 



SUDDENLY APPLIED STRESSES 



[Chap. I 



18. Stresses Due to Suddenly Applied Forces. — In studying 
the stresses produced by suddenly applied forces, two distinct 
cases must be considered. 

(a) An unstrained member acted upon by a suddenly applied 
force having no velocity of approach. 

(6) An unstrained member acted upon by a force that has a 
velocity of approach. 

Case (a) . — For the case in which the suddenly applied force P 
has no velocity before striking the unstrained member, the exter- 
nal work done by this force is PA, in which A 
represents the total deformation of the mem- 
ber. If the stress S induced in the member 
having an area A does not exceed the elastic 
limit, then the internal work is represented 
by the following expression: 



lip 



w 

7 



Internal work = 



AAS 



TEED 

Fig. 5. 



Equating the external to the internal work, 
we obtain 

9 P 

S = Z -f (31) 



That is, the stress produced in this case by 
the suddenly applied force P is double that 
produced by the same force if it were applied 
gradually. 

Case (6) . — To derive the expression for the 
magnitude of the stress induced in an un- 
strained member of area A by a force P that 
has a velocity of approach v, we shall assume a long bolt or bar 
having a head at one end and the other end held rigidly as 
shown in Fig. 5. Upon the bolt a weight W slides freely, and 
is allowed to fall through a distance b before it strikes the head 
of the bolt. 

As soon as the weight W strikes the head, the bolt will elongate 
a distance A, from which it is evident that the external work 
performed by W is W(b + A). The stress in the bolt at the 
instant before W strikes the head is zero, and after the bolt has 
been elongated a distance A the stress is S; hence the work of the 

variable tension during the period of elongation is , assuming 



Art. 19] REPEATED STRESSES 19 

that S is within the elastic limit. To do this internal work, the 
weight W has given up its energy; hence equating the external 
to the internal work and solving for S, we get 

2W 

S = ~(b + A) (32) 

From Art. 5, the elongation 

Substituting this value of A in (32), and collecting terms 



s = ^i + Ji+ 2bAE 



WL 



(33) 



If in (33), the distance b is made zero, so as to give the condi- 

2 W 
tions stated in case (a) above, we find that S = ~~a~, which agrees 

with results expressed by (31). 

REPEATED STRESSES 

19. Repeated High Stresses. — It is now generally conceded 
that in a machine part subjected to repeated stress there is some 
internal wear or structural damage of the material which eventu- 
ally causes failure of the part. In June, 1915, Messrs. Moore 
and Seely presented before the American Society for Testing 
Materials a paper, in which they gave an excellent analytical 
discussion of the cumulative damage done by repeated stress. 
The application of the proposed formula gives results that agree 
very closely with the experimental results obtained by the authors 
themselves as well as those obtained by earlier investigators. For 
a range of stress extending from the yield point to a stress slightly 
below the elastic limit, Messrs. Moore and Seely derived the 
following formula as representing the relation existing between 
the fiber stress and the number of repetitions of stress necessary 
to cause failure: 

s = (T^W ^ 

in which 

N = the number of repetitions of stress. 

S = maximum applied unit stress (endurance strength). 

a = constant depending upon the material. 



20 



REPEATED STRESSES 



[Chap. I 



b = constant based upon experiment. 

_ minimum unit stress^ 
y maximum unit stress 

For a complete reversal of stress, q = — 1, and when the range 
is from zero to a maximum, q = 0. 

20. Repeated Low Stresses. — The formula expressing the re- 
lation between the fiber stress and the number of repetitions of 
low stress, according to the above-mentioned paper, is as follows : 



S = 



(1 + cN e ), 



(35) 



(1 - q)N b 

in which c and e are constants, the values of which must be ob- 
tained by means of experiments. The factor (1 + cN e ) is called 



Number of Repetitions 
5xl0 9 I0 9 ' 5i 









































1.20 z 

hi 5 4. 

o 

> 








■ — L^ 












k 


\ 


































































































































= 









I0 7 5xl0 7 

Number of Repetitions 
Fig. 6. 



108 



by the authors a probability factor, and its numerical value de- 
pends altogether upon the judgment of the designer. In Fig. 6 
are plotted the values of (1 + cN e ), as proposed by the authors, 
for use in determining the magnitude of the stress S in any part, 
the failure of which would not endanger life. For parts, the fail- 
ure of which would endanger life, this probability factor should 
be assumed as -equal to unity. In Table 3 are given values of 
a for various materials, as determined from existing data of 
repeated stress tests. 



Art. 21] 



SAFE ENDURANCE STRESS 
Table 3. — Values of Constant 



21 



Material 


a 


Material 


a 




250,000 

250,000 
400,000 
350,000 
250,000 


Spring steel 

Hard-steel wire .... 

Gray cast iron 

Cast aluminum .... 
Hard-drawn copper 
wire. 


400,000 to 
600,000 
600,000 
100,000 
80,000 
140,000 


Soft machinery steel 

Cold-rolled steel shafting . . . 
Steel (0.45 per cent, carbon) 
Wrought iron 



The value of q, the ratio of minimum to maximum stress is 
usually known, or may be established from the given data. Ac- 
cording to the authors, if the stress is wholly or partially reversed, 
q must be taken as negative, having a value of — 1 when there 
is a complete reversal of stress. In cases where the value of q 
approaches +1, it is possible that the endurance stress calcu- 
lated by means of (35), will be in excess of the safe static stress, in 
which case the latter should govern the design. 

For the exponent b, Messrs. Moore and Seely recommend that 
it should be made equal to %, this value being derived from a 
careful study of data covering a wide range of repeated stress 
tests. 

21. Safe Endurance Stress. — As stated in a preceding para- 
graph, the formula given applies only to stresses up to the yield 
point of the material; hence whenever the endurance strength 
calculated by (35) is less than the yield point, a so-called factor 
of safety must be introduced, in order to arrive at a safe endurance 
stress. This may be accomplished in the following two ways: 

(a) By applying the factor of safety to the stress. 

(6) By applying the factor of safety to the number of repeti- 
tions. 

The latter method is recommended by Moore and Seely, and 
the method of procedure is to multiply the number of repetitions 
a machine is to withstand by the factor of safety, and then deter- 
mine the endurance stress for this new number of repetitions. 



TEMPERATURE STRESSES 



22. Deformation Due to Temperature Change.— It is important 
that certain machine members be so designed that expansion as 



22 



TEMPERATURE STRESSES 



[Chap. 1 



well as contraction due to a change in temperature may take place 
without unduly stressing the material. Now before we can de- 
termine the magnitude of such stresses, we must arrive at the 
deformation caused by the rise or drop in temperature. The 
amount that a member will change in length depends upon the 
material and the change in temperature, and may be expressed by 
the following formula: 

A = cdL, (36) 



in which L represents the original length, t the change in tempera- 
ture in degrees Fahren- 
Table 4. — Values op Coefficient of 
Linear Expansion 



Material 



Range of 
temperature 



Coefficient 



Cast iron. 



Wrought iron < 

Steel casting. . . 

Soft steel 

Nickel steel... . 
Brass casting. . 
Bronze 

Copper 



heit, and a the coefficient 
of linear expansion. For 
values of a consult Table 4. 

23. Stress Due to Tem- 
perature Change.— Due 

to the deformation A dis- 
cussed in the preceding 
article, the machine mem- 
ber subjected to a change 
in temperature will be 
stressed, if its ends are 
constrained so that no 
expansion or contraction 
may occur. Knowing the 

magnitude of A, the unit strain is j> from which we may 

readily determine the intensity of stress due to a change t in 
temperature, by applying the definition of the modulus of elas- 
ticity given in Art. 5; hence 

S = atE (37) 



32 to 212 
32 to 212 
32 to 572 
32 to 212 
32 to 212 
32 to 212 
32 to 212 
32 to 212 
32 to 212 
32 to 572 



0.00000618 

0.00000656 

0.00000895 

0.00000600 

0.00000630 

0.00000730 

0.0000104 

0.0000100 

0.00000955 

0.00001092 



WORKING STRESSES 

24. Factor of Safety. — In general, the maximum stress in- 
duced in a machine part must be kept well within the elastic 
limit so that the action of the external forces is almost perfectly 
elastic. The stress thus used in arriving at the size of the part 
is called the working stress, and its magnitude depends upon the 
following conditions: 

(a) Is the application of load steady or variable? 



Art. 24] WORKING STRESSES 23 

(6) Is the part subjected to unavoidable shocks or jars? 
.(c) Kind of material, whether cast iron, steel, etc. 

(d) Is the material used in the construction reliable? 

(e) Is human life or property endangered, in case any part of 
a machine fails? 

(/) In case of failure of any part, will any of the remaining 
parts of the machine be overloaded? 

(g) Is the material of the machine part subjected to unneces- 
sary and speedy deterioration? 

(h) Cost of manufacturing. 

(i) The demand upon the machine at some future time. 

As usually determined, the working stress for a given case is 
obtained by dividing the ultimate strength by the so-called 
factor of safety, which factor should really represent a product of 
several factors depending upon the various conditions enumerated 
above. In general, larger factors of safety are used when a piece 
is made of cast metal, than when a hammered or rolled material 
is used. The selection of a larger factor of safety for cast metals 
is due to the fact that cast parts may contain hidden blow holes 
and spongy places. In many cases the material may be stressed 
an unknown amount due to unequal cooling caused by the im- 
proper distribution of the material, no matter how careful the 
moulder may be in cooling 
the casting after it is poured. Table 5.— Suggested Factors of 

Again, live loads require 
much larger factors of safety 
than dead loads, and loads 
that produce repetitive 
stresses that change con- 
tinually from tension to com- 
pression, for example, also 
require large factors of 
safety, the magnitudes of 
which are difficult to deter- 
mine. For the latter case, the equations of Arts. 19, 20 and 
21 may serve as guide. 

In Table 5 are given suggested factors of safety based on the 
ultimate strength of the material. It must be remembered that 
the skill and judgment of the designer should play an important 
part in arriving at the proper working stresses for any given set 
of conditions. 





Kind of stress 


Material 


Steady 


Varying 


■ Shock 


Hard steel 


5 


6 


15 


Structural steel. 


4 


6 


10 


Wrought iron . . 


4 


6 


10 


Cast iron 


6 


10 


20 


Timber 


6 


10 


15 



24 



TABLE OF PHYSICAL PROPERTIES 



[Chap. I 



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Art. 24] REFERENCES 25 

For ultimate strengths and various other physical properties 
of the more common metals used in the construction of machinery, 
consult Table 6. 

References 

Mechanics of Materials, by Merriman. 
The Strength of Materials, by E. S. Andrews. 
Mechanical Engineers' Handbook, by L. S. Marks. 
Elasticitat und Festigkeit, by C. Bach. 



CHAPTER II 

MATERIALS USED IN THE CONSTRUCTION OF 
MACHINE PARTS 

The principal materials used in the construction of machine 
parts are cast iron, malleable iron, steel casting, steel, wrought 
iron, copper, brass, bronze, aluminum, babbitt metal, wood and 
leather. 

CAST IRON 

25. Cast Iron. — Cast iron is more commonly used than any 
other material in making machine parts. This is because of its 
high compressive strength and because it can be given easily 
any desired form. A wood or metal pattern of the piece desired 
is made, and from this a mould is made in the sand. The pattern 
is next removed from the mould and the liquid metal is poured 
in, which on cooling assumes the form of the pattern. 

Crude cast iron is obtained directly from the melting of the 
iron ore in the blast furnace. This product is then known as 
pig iron, and is rarely ever used except to be remelted into cast 
iron, or to be converted into wrought iron or steel. Cast iron 
fuses easily, but it cannot be tempered nor welded under ordinary 
conditions. The composition of cast iron varies considerably, 
but in general is about as follows : 

Per cent. 

Metallic iron 90.0 to 95.0 

Carbon 1.5 to 4.5 

Silicon 0.5 to 4.0 

Sulphur less than 0. 15 

Phosphorus 0. 06 to 1 . 50 

Manganese .• . . . trace to 5.0 

(a) Carbon. — 'Carbon may either be united chemically with 
the iron, in which case the product is known as white iron, or it 
may exist in the free state, when the product is known as gray 
iron. The white iron is very brittle and hard, and is therefore 
butjittle used in machine parts. In the free state the carbon 
exists as graphite. 

26 



Art. 26] CAST IRON 27 

(b) Silicon. — Silicon is an important constituent of cast iron 
because of the influence it exerts on the condition of the carbon 
present in the iron. The presence of from 0.25 to 1.75 per cent, 
of silicon tends to make the iron soft and strong; but beyond 2.0 
per cent, silicon, the iron becomes weak and hard. An increase 
of silicon causes less shrinkage in the castings, but a further in- 
crease (above 5 per cent.) may cause an increase in the shrinkage. 
With about 1.0 per cent, silicon, the tendency to produce blow 
holes in the castings is reduced to a minimum. 

(c) Sulphur. — Sulphur in cast iron causes the carbon to unite 
chemically with the iron, thus producing hard white iron, which 
is brittle. For good castings, the sulphur content should not 
exceed 0.15 per cent. 

(d) Phosphorus. — Phosphorus in cast iron tends to produce 
weak and brittle castings. It also causes the metal to be very 
fluid when melted, thus producing an excellent impression of the 
mould. For this reason phosphorus is a desirable constituent 
in cast iron for the production of fine, thin castings where no 
great strength is required. To produce such castings, from 2 to 5 
per cent, of phosphorus may be used. For strong castings of 
good quality, the amount of phosphorus rarely exceeds 0.55 per 
cent., but when fluidity and softness are more important than 
strength, from 1 to 1.5 per cent, may be used. 

(e) Manganese. — Manganese when present in cast iron up to 
about 1.5 per cent, tends to make the castings harder to machine; 
but renders them more suitable for smooth or polished surfaces. 
It also causes a fine granular structure in the castings and pre- 
vents the absorption of the sulphur during melting. Man- 
ganese may also be added to cast iron to soften the metal. This 
softening is due to the fact that the manganese counteracts the 
effects of the sulphur and silicon by eliminating the former and 
counteracting the latter. However, when the iron is remelted, 
its hardness returns since the manganese is oxidized and more 
sulphur is absorbed. The transverse strength of cast iron is 
increased about 30 per cent., and the shrinkage and depth of 
chill decreased 25 per cent., while the combined carbon is dimin- 
ished one-half by adding to the molten metal, powdered ferro- 
manganese in the proportion of 1 pound of the latter to about 
600 pounds of the former. 

26. Vanadium Cast Iron. — The relatively coarse texture of 
cast iron may be much improved by the addition of 0.10 to 0.20 



28 



PIG IRON 



[Chap. II 



per cent, of vanadium, and at the same time the ultimate strength 
is increased from 10 to 25 per cent. Cast iron containing a small 
percentage of vanadium is tougher than ordinary gray iron, thus 
making it an excellent material for use in steam- and gas-engine 
cylinders, piston rings, liners, gears and other similar uses. Some 
of the larger railway systems have now adopted this material for 
their cylinder construction. In machining vanadium cast iron, 
it is possible to give it a much higher finish than is possible with 
gray iron. 

27. Pig Iron. — Pig iron is the basis for the manufacture of all 
iron products. It is not only used practically unchanged to pro- 
duce castings of a great variety of form and quality, but it is also 
used in the manufacture of wrought iron and steel. For each 
special purpose, the iron must have a composition within certain 
limits. It follows, therefore, that pig iron offers a considerable 
variety of composition. The practice of purchasing pig iron 
by analysis is generally followed at the present time. In Table 
7 are give the specifications for the various grades of pig iron 
used by one large manufacturer. 

Table 7. — Specifications op Pig Iron 



Class 


Total carbon 
not under, 
per cent. 


Silicon, 
per cent. 


Sulphur 
not over, 
per cent. 


Phosphorus, 
per cent. 


Manganese 
not over, 
per cent. 


1 


3.0 


1 . 5 to 2 . 


0.040 


0.20 to 0.75 


1.0 


2 


3.5 


2 . to 2 . 5 


0.035 


0.20 to 0.75 


1.0 


3 


3.5 


2 . 5 to 3 . 


0.030 


0.20 to 0.75 


1.0 


4 


3.5 


2.0to2.5 


0.040 


1.00 to 1.50 


1.0 


5 


3.0 


4.0to5.0 


0.040 


0.20 to 0.80 


l.p 



In general, an analysis is made from drillings taken from a pig 
selected at random from each four tons of every carload as un- 
loaded. The right is reserved to reject a portion or all of the 
material which does not conform to the above specifications in 
every particular. 

In a general way, the specified limits for the composition of the 
chief grades of pig iron are given in Table 8. 

According to use, pig iron may be divided roughly into two 
classes. The first class includes those grades used in the produc- 
tion of foundry and malleable irons, while the second includes 
those used in the manufacture of wrought iron and steel. In the 
process of remelting or manufacturing, the first class undergoes 



Art. 28] 



CHILLED CASTING 



29 



little if any chemical change, while the second class undergoes 
a complete chemical change. 

Table 8. — General Specifications of Pig Iron 



Grade of iron 



Silicon, 
per cent. 



Sulphur, 
per cent. 



Phosphorus, 
per cent. 



Manganese, 
per cent. 



No. 1 foundry.. . 
No. 2 foundry.. . 
No. 3 foundry.. . 

Malleable 

Gray forge 

Bessemer 

Low phosphorus 

Basic 

Basic Bessemer.. 



2.5 to3.0 
2.0to2.5 
1.5to2.0 
0.7tol.5 
Under 1 . 5 
1.0to2.0 
Under 2 . 
Under 1 . 
Under 1 . 



Under 
Under 
Under 
Under 
Under 
Under 
Under 
Under 
Under 



0.035 
0.045 
0.055 
0.050 
0.100 
0.050 
0.030 
0.050 
0.050 



0.5 to 1.0 

Under 0.2 
Under 1 . 
Under 0.1 
Under 0.3 
Under 1 . 
2 . to 3 . 



Under 1 . 



1.0to2.0 



28. Malleable Casting. — Malleable castings are made by heat- 
ing clean foundry castings, preferably with the sulphur content 
low, in an annealing furnace in contact with some substance that 
will absorb the carbon from the cast iron. Hematite or brown 
iron ore in pulverized form is used extensively for that purpose. 
The intensity of heat required is on the average about 1,650°F. 
The length of time the castings remain in the furnace depends 
upon the degree of malleability required and upon the size of the 
castings. Usually light castings require a minimum of 60 hours, 
while the heavier ones may require 72 hours or longer. 

The tensile strength of good malleable cast iron lies somewhere 
between that of gray iron and steel, while its compressive stress 
is somewhat lower than that of the former. Good malleable cast- 
ings may be bent and twisted without showing signs of fracture, 
and for that reason are well adapted for use in connection with 
agricultural machinery, railroad supplies, and automobile parts. 

29. Chilled Casting. — Chilled castings are those which have a 
hard and durable surface. The iron used is generally close-grained 
gray iron low in silicon. A chilled casting is formed by making 
that part of the mould in contact with the surface of the casting 
to be chilled of such construction that the heat will be with- 
drawn rapidly. The mould for causing the chill usually con- 
sists of iron bars or plates, placed so that their surfaces will be in 
contact with the molten iron. These plates abstract heat rapidly 
from the iron, with the result that the part of the casting in con- 



30 WROUGHT IRON [Chap. II 

tact with the cold surface assumes a state similar to white iron, 
while the rest of the casting remains in the form of gray iron. 
The withdrawal of heat is hastened by the circulation of cold 
water through pipes, circular or rectangular in cross-section, placed 
near the surface to be chilled. Chilled castings offer great re- 
sistance to crushing forces. The outside or "skin" of the ordi- 
nary casting is in fact a chilled surface, but by the arrangement 
mentioned above, the depth of the "skin" is greatly increased 
with a corresponding increase in strength and wearing qualities. 
Car wheels, jaws for crushing machinery, and rolls for rolling 
mills are familiar examples of chilled castings. Car wheels re- 
quire great strength combined with a hard durable tread. The 
depth of the chill varies from % to 1 inch. 

It has been found that with the use of vanadium in chilled 
castings, a deeper, stronger and tougher chill can be produced. 
This chill, however, is not quite as hard as that found on ordinary 
chilled cast iron, and hence has the advantage that such castings 
can be filed and machined more easily. 

30. Semi-steel. — The term semi-steel is applied to a metal that 
is intermediate between cast iron and malleable iron. The 
meaning of the term as used at the present time is vague and for 
that reason its use is questioned. The so-called semi-steel is 
produced in the cupola by mixing from 20 to 40 per cent, of low- 
carbon steel scrap with the pig iron and cast scrap. This mix- 
ture, if properly handled in the cupola as well as in pouring the 
mould, produces a clean close-grained tough casting that may be 
machined easily and that has an ultimate tensile strength vary- 
ing from 32,000 to 42,000 pounds per square inch. Its trans- 
verse strength is also considerably higher than that of ordinary 
gray iron. However, the material produced by such a mixture 
as given above has none of the distinctive properties of steel and 
in reality it is nothing more than a high-grade gray-iron casting. 
Semi-steel has been used very successfully for cylinders, piston 
rings, cylinder liners, gears, plow points, and frames of punches 
and shears. 

WROUGHT IRON 

31. Wrought Iron. — Wrought iron is formed from pig iron by 
melting the latter in a puddling furnace. During the process of 
melting, the impurities in the pig iron are removed by oxidation, 
leaving the pure iron and slag both in a pasty condition. In this 



Art. 32] STEEL CASTING 31 

condition the mixture of iron and slag is formed into muck balls 
weighing about 150 pounds, and is removed from the furnace 
These balls are put into a squeezer and compressed, thereby re- 
moving a large amount of the slag, after which it is rolled into 
bars. The bars, known as "muck bars," are cut into strips and 
arranged in piles, the strips in the consecutive layers being at 
right angles to each other. These piles are then placed into a 
furnace and raised to a welding heat and are then rolled into mer- 
chant bars. If the quality of the iron is to be improved and the 
last-mentioned process is repeated, we obtain what is known as 
"best iron," "double best" and "treble best," depending upon the 
number of repetitions. The merchant bar finally produced is 
the ordinary wrought iron of commerce. At the present time 
wrought iron is not used as extensively as in the past, steel to a 
great extent having taken its place ; however, it still is used in the 
manufacture of pipes, boiler tubes, forgings, parts of electrical 
machinery, small structural shapes, and crucible steel. 

STEEL CASTING 

32. Manufacturing Processes. — Castings similar to iron cast- 
ings may be formed in almost any desired shape from molten 
steel. They are produced by four distinct methods as follows: 

(a) Crucible process. — When it is desired to produce very fine 
and high-grade castings, not very large, the crucible process is used. 

(6) Bessemer process. — This method is used chiefly for pro- 
ducing small castings. 

(c) Open-hearth process. — The open-hearth process is used ex- 
tensively for the production of steel castings either small or ex- 
tremely large in size. The castings produced by this method are 
considered superior to those produced by the Bessemer process. 

(d) Electric-furnace method. — The electric furnace which is now 
being introduced into this country is capable of producing the 
very best grades of steel castings. 

In texture, the castings produced by the common processes in 
use today are coarse and crystalline, since the steel has been per- 
mitted to cool without drawing or rolling. In order to improve 
the grain structure, and at the same time remove some of the 
internal stresses, all steel castings must be annealed before ma- 
chining them. Formerly trouble was experienced in obtaining 
good sound steel castings ; but by great care and improved meth- 



32 MANGANESE-STEEL CASTINGS [Chap. II 

ods in the production of moulds, first-class castings may now 
be obtained. In general, steel castings are used for those machine 
parts requiring greater strength than is obtained by using gray- 
iron castings. 

33. Manganese -steel Castings. — Manganese-steel castings are 
produced by adding ferro-manganese to open-hearth steel, and the 
average chemical composition of such castings is about as follows : 
Manganese, 12.5 per cent.; carbon, 1.25 per cent.; silicon, 0.3 per 
cent.; phosphorus, 0.08 per cent.; sulphur, 0.02 per cent.; iron, 
85.85 per cent. The average physical properties of this kind of 
steel casting are about as follows: 

Tensile strength 110,000 pounds per square inch. 

Elastic limit 54,000 pounds per square inch. 

Elongation in 8 inches. ... 45 per cent. 

Reduction of area 50 per cent. 

Manganese steel is in general free from blow holes, but is diffi- 
cult to cast on account of its high shrinkage, which is about two 
and one-half times as great as that of cast iron. As originally 
cast it is extremely hard and brittle and it is possible to pulverize 
it under the blows of a hammer. The fact that this metal is 
brittle when it comes from the mould makes it possible to break 
off the risers and .gates remaining on the casting, which could 
not be done were the original casting as tough as the finished 
product. As mentioned, manganese-steel casting possesses great 
hardness which is not diminished by annealing, and in addition 
it has a high tensile strength combined with great toughness and 
ductility. These qualities would make this steel the ideal metal 
for machine construction, were it not for the fact that its great 
hardness prevents it from being machined in any way but by 
abrasive processes, which at best are expensive. Again, the 
very property of hardness, combined with great toughness, also 
limits its use to the rougher class of castings, or such that require 
a minimum amount of finish. 

The toughness of the finished casting is produced by the an- 
nealing process. In this process the brittle castings are placed 
in annealing furnaces, in which they are heated gradually and 
carefully. After remaining in these furnaces from three to 
twenty-four hours, depending upon the type of casting treated, 
the castings are removed from the furnace and quenched in cold 
water. It is evident that great care must be exercised by the 



Art. 34] MANGANESE-STEEL CASTINGS 33 

designer to distribute the metal properly in large and complicated 
castings in order that all the parts may cool at approximately 
the same rate. It has been found by experience that the heat 
treatment just described cannot be made to extend through a 
section thicker than 5 or 5% inches. In general, thicknesses 
exceeding 3^ inches are not found in well-designed casting. 

34. Applications of Manganese-steel Castings. — Due to the 

fact that manganese-steel casting is the most durable metal known 
as regards ability to resist wear, it is well adapted to the following 
classes of service : 

(a) For all wearing parts of crushing and pulverizing machinery, 
such as rolls, jaws and toggle plates, heads, mantels, and concaves. 

(&) In all classes of excavating machinery; for example, the 
dipper and teeth of dipper dredges, the buckets of placer dredges, 
the cutter head and knives of ditching machines. 

(c) The impellers and casings of centrifugal pumps are fre- 
quently made of manganese-steel casting. In this connection it 
is of interest to note that soft-steel inserts are cast into the casing 
at proper places to permit the drilling and tapping of holes for 
the various attachments. 

(d) In connection with hoisting machinery such parts as sheaves, 
drums, rollers, and crane wheels made of manganese-steel cast- 
ing are not uncommon. It is claimed that the life of a rope 
sheave or roller made of this material is about thirty times that 
of one made of cast iron. 

(e) In mining work, the wheels of coal cars and skips, also the 
head sheaves, are made of manganese steel. In the latter appli- 
cation, the rim only is made of manganese steel and is then 
bolted to the wrought-iron spokes, which in turn are bolted to 
the cast-iron hub. 

(/) In conveying machinery where the parts are subjected to 
severe usage, as for example a conveyor chain in a cement mill, 
both the chains and the sprockets are made of manganese-steel 
casting. 

(g) In railway track work, manganese-steel casting has given 
excellent service for crossings, frogs, switches, and guard rails. 

(h) Another very important use of manganese-steel casting 
is in the construction of safes and vaults; for this purpose it is 
particularly well adapted since it cannot be drilled nor can its 
temper be drawn by heating. 



34 BESSEMER PROCESS [Chap. II 

STEEL 

Steel is a compound in which iron and carbon are the principal 
parts. It is made from pig iron by burning out the carbon, 
silicon, manganese and other impurities, and recarbonizing to 
any degree desired. The principal processes or methods of 
manufacturing steel are the following: (a) the Bessemer; (b) 
the open-hearth; (c) the cementation. 

35. Bessemer Process. — In the Bessemer process, several tons, 
usually about ten, of molten pig iron are poured into a con- 
verter, and through this mass of iron a large quantity of cold air 
is passed. In about four minutes after the blast is turned on, 
all the silicon and manganese of the pig iron has combined with 
the oxygen of the air. The carbon in the pig iron now begins 
to unite with the oxygen, forming carbon monoxide, which burns 
through the mouth of the converter in a long brilliant flame. 
The burning of the carbon monoxide continues for about six 
minutes, when the flame shortens, thus indicating that nearly 
all of the carbon has been burned out of the iron and that the 
air supply should be shut off. The burning out of these impuri- 
ties has raised the temperature of the iron to a white heat, and 
at the same time produced a relatively pure mass of iron. To 
this mass is added a certain amount of carbon in the form of a 
very pure iron high in carbon and manganese. The metal is 
then poured into moulds forming ingots, which while hot are 
rolled into the desired shapes. 

The characteristics of the Bessemer process are: (a) great 
rapidity of reduction, about ten minutes per heat; (6) no extra 
fuel is required; (c) the metal is not melted in the furnace where 
the reduction takes place. 

Bessemer steel was formerly used almost entirely in the manu- 
facture of wire, skelps for tubing, wire nails, shafting, machinery 
steel, tank plates, rails, and structural shapes. Open-hearth 
steel, however, has very largely superseded the Bessemer product 
in the manufacture of these articles. 

36. Open -hearth Process.— In the manufacture of open-hearth 
steel, the molten pig iron, direct from the reducing furnace, is 
poured into a long hearth, the top of which has a firebrick lining. 
The impurities in the iron are burned out by the heat obtained 
from burning gas and air, and reflected from this refractory lining. 



Art. 37] CRUCIBLE STEEL 35 

The slag is first burned, and the slag in turn oxidizes the im- 
purities. The time required for purifying is from 6 to 10 hours, 
after which the metal is recarbonized, cast into ingots and rolled 
as in the Bessemer process. 

The characteristics of the open-hearth process are : (a) relatively 
long time to oxidize the impurities; (6) large quantities, 35 to 
70 tons, may be purified and recarbonized in one charge; (c) 
extra fuel is required; (d) a part of the charge, steel scrap and 
iron ore added at the beginning of the process, are melted in the 
furnace. 

Open-hearth steel is used in the manufacture of cutlery, boiler 
plate, and armor plate in addition to the articles mentioned in 
Art. 35. 

37. Cementation Process. — In this process of manufacturing 
steel, bars of wrought iron imbedded in charcoal are heated for 
several days. The wrought iron absorbs carbon from the char- 
coal and is thus transformed into steel. When the bars of iron 
are removed they are found to be covered with scales or blisters. 
The name given to this product is blister steel. By removing the 
scales and blisters and subjecting the bars to a cherry-red heat 
for a few days, a more uniform distribution of the carbon is 
obtained. 

Blister steel when heated and rolled directly into the finished 
bars, is known as German steel. Bars of blister steel may be cut 
up and forged together under the hammer, forming a product 
called shear steel. By repeating the process with the shear steel, 
we obtain double-shear steel. 

38. Crucible Steel. — Crucible steel, also called cast steel, is 
very uniform and homogeneous in structure. It is made by 
melting blister steel in a crucible, casting it into ingots and rolling 
into bars. By this method is produced the finest crucible steel. 
Another method of producing crucible-cast steel is to melt 
Swedish iron (wrought iron obtained from the reduction of a 
very pure iron in the blast furnace in which charcoal instead of 
coke for producing the puddling flame is used) in contact with 
charcoal in a sealed vessel, the contents of which are poured 
into a large ladle containing a similar product from other sealed 
vessels. This mixing insures greater uniformity of material. 
The metal in this large ladle is cast into ingots, which are sub- 



36 NICKEL STEEL [Chap. II 

sequently forged or rolled into bars. By far the greater part 
of crucible steel is produced by this method. 

39. Cold-rolled Steel. — The so-called cold-rolled steel is rolled 
hot to approximately the required dimensions. The surface is 
then carefully cleaned, usually by chemical means, and rolled 
cold to a very accurately gauged thickness between smooth 
rollers. The rolling of metal when cold has two important 
advantages as follows: when steel is rolled hot the surface of 
the steel oxidizes and forms a scale, while with cold rolling no 
such action takes place, thus making it possible to produce a 
bright finish. Furthermore, since no scale is formed the bar or 
plate to be rolled can be made very accurate. The cold-rolling 
process has the effect of increasing the elastic limit and ultimate 
strength, but decreases the ductility. It also produces a very 
smooth and hard surface. Its principal use is for shafting and 
rectangular, square and hexagonal bars, as well as strip steel 
which of late is in demand for use in the manufacture of pressed- 
steel products. For the latter class of work the absence of scale, 
already referred to, has a marked effect on the life of the dies, 
as experience in press working of hot-rolled metal shows that the 
scale on the latter is exceedingly hard on the dies. 

ALLOY STEELS 

The term alloy steels is applied to all steels that are composed 
of iron and carbon, and one or more special elements such as 
nickel, tungsten, manganese, silicon, chromium, and vanadium. 
In general, alloy steels must always be heat treated, and should 
never be used in the natural or annealed condition, since in the 
latter condition the physical properties of the material are but 
little better than those of the ordinary carbon steels. The heat 
treatment given to alloy steels causes a marked improvement in 
the physical properties. A few of the principal alloy steels are 
discussed in the following paragraphs. 

40. Nickel Steel. — Nickel added to a carbon steel increases its 
ultimate strength and elastic limit as well as its hardness and 
toughness. It tends to produce a steel that is more homogeneous 
and of finer structure than the ordinary carbon steel, and if the 
percentage of nickel is considerable the material produced resists 
corrosion to a remarkable degree. The percentage of nickel 



Aht. 41] VANADIUM STEEL 37 

varies from 1.5 to 4.5, while the carbon varies from 0.15 to 0.50 
per cent., both of these percentages depending upon the grade 
of steel desired. Nickel steel has a high ratio of elastic limit to 
ultimate strength and in addition offers great resistance to crack- 
ing. The latter property makes this type of steel desirable for 
use as armor plate. Nickel steel is also used for structural 
shapes and for rails; the latter show better wearing qualities 
than those made from Bessemer or open-hearth steel. On 
account of its ability to withstand heavy shocks and torsional 
stresses, nickel steel is well adapted for crankshafts, high-grade 
shafting, connecting rods, automobile parts, car axles and 
ordnance. 

41. Chrome Steel. — Chrome steel is produced by adding to 
high-carbon steel (0.8 to 2.0 per cent.) from 1 to 2 per cent, of 
chromium. The steel thus produced is very fine-grained and 
homogeneous, is extremely hard, and has a high ratio of elastic 
limit to ultimate strength. Due to its extreme hardness, chrome 
steel may be used for ball and roller bearings, armor-piercing 
shells, armor plate, burglar-proof safes, and vaults. The element 
chromium is also used in the manufacture of the best high-speed 
tool steels. 

42. Vanadium Steel. — Vanadium steel is produced by adding 
to carbon steel, a small amount of vanadium, generally between 
0.15 and 0.25 per cent. This alloy steel is used as a forging or 
machinery steel, and should be heated slowly when preparing 
it for a forging operation. The effect of the vanadium is to 
increase the elastic limit as well as the capacity for resisting shock. 
Vanadium is used more in conjunction with chromium or nickel 
steel than with ordinary carbon steel. Carbon vanadium steel 
containing from 0.60 to 1.25 per cent, carbon and over 0.2 per 
cent, vanadium may be tempered, and due to its toughness, is 
well adapted for punches, dies, rock drills, ball and roller bearings, 
and other similar uses. 

43. Nickel-chromium Steel. — Nickel-chromium steel is used 
chiefly in automobile construction, where a high degree of strength 
and hardness is demanded. At the present time this type of 
steel is also being used for important gears on machine tools. 
In the automobile industry, three types of nickel-chromium steels 
are commonly used. These are known as low nickel-, medium 
nickel-, and high nickel-chromium steels. 



38 CHROMIUM-VANADIUM STEEL [Chap. II 

In general, nickel-chromium steels having a carbon content 
up to 0.2 per cent, are intended for case hardening; those having 
0.25 to 0.4 per cent, carbon are used for the structural parts of 
automobiles, while the higher-carbon steels may be used for gears 
or other important parts. 

44. Chromium-vanadium Steel.— Chromium-vanadium steel 
is tough and capable of resisting severe shocks, and has an exceed- 
ingly high elastic limit in proportion to its ultimate strength. 
This type of steel is used for springs, gears, driving shafts, steering 
knuckles, and axles in the automobile industry. It is also used 
for spindles and arbors for machine tools, locomotive driving 
axles, piston rods, side and connecting rods, and locomotive and 
car-wheel tires. 

For high-duty shafts requiring a high degree of strength and a 
moderate degree of toughness, the grade of chromium-vanadium 
steel containing about 0.4 per cent, carbon should be selected. 
For springs and gears the carbon content should be from 0.45 to 
0.50 per cent. Chromium-vanadium steels having a high carbon 
content of 0.75 to 1.0 per cent, may be tempered and used for 
tools. In addition to being hard it is tough, and for that reason 
has been used successfully for dies, punches, ball-bearing races, 
rock drills, and saws. 

45. Silicon-manganese Steel. — A combination of silicon and 
manganese in moderate amounts added to steel increases its 
capacity for resisting shock, thus making it particularly suitable 
for all kinds of springs and to some extent for gears. For each 
class of service mentioned the steel must be given a proper heat 
treatment. 

46. Tungsten Steel. — Tungsten steel is an alloy of iron, carbon, 
tungsten and manganese, and sometimes chromium. The- ele- 
ment which gives this steel its peculiar property, self or air harden- 
ing, is not tungsten but manganese combined with carbon. The 
tungsten, however, is an important element, since it enables the 
alloy to contain a larger percentage of carbon. On account of its 
hardness, this steel can not be easily machined, but must be 
forged to the desired shape. Its chief use is for high-speed cutting 
tools. 

ALLOYS 

Alloys may be made of two or more metals that have an affinity 
for each other. The compound or alloy thus produced has 



Art. 47] BRASS ■ 39 

properties and characteristics which none of the metals possess. 
The principal alloys used in machine construction may be ob- 
tained by combining two or more of the following metals : copper, 
zinc, tin, lead, antimony, bismuth, and aluminum. 

47. Brass. — Brass is an alloy of copper and zinc; however, 
many of the commercial brasses contain small percentages of 
lead, tin, and iron. Brass for machine parts may be put in two 
general classes, namely, cast brass and wrought brass. 

(a) Cast brass. — Cast brass is intended for parts not requiring 
great strength, and as usually made has a zinc content of about 
35 per cent., and the remainder copper with traces of iron, lead 
and tin. In order to make cast brass free-cutting for machining 
purposes 1 to 2 per cent, of lead is added. A typical specification 
for cast brass as used by the Bureau of Steam Engineering of the 
United States Navy Department is as follows: copper, 59 to 63 
per cent.; tin, 0.5 to 1.5 per cent.; iron, not exceed 0.06 per cent.; 
lead, not exceed 0.60 per cent.; zinc, remainder. 

(6) Wrought brass. — Wrought brass may be of two kinds as 
follows: 1. That which contains approximately 56 to 62 per 
cent, of copper and the remainder zinc may be rolled or forged 
while hot. Muntz metal containing 60 per cent, of copper and 
40 per cent, zinc is a well-known wrought brass which at one time 
was used very extensively for ship sheathing. The so-called 
Tobin bronze is another type of wrought brass that may be 
worked while hot, but it differs from Muntz metal in that it 
contains very small percentages of iron, tin, and lead, in addition 
to the copper and zinc. Its ultimate tensile strength is about 
equal to that of ordinary steel, while its compressive strength is 
about three times its tensile strength. Tobin bronze resists 
corrosion and for that reason meets with favor in naval work. 

2. The second kind of wrought brass contains approximately 
70 per cent, of copper and 30 per cent, of zinc, and not infre- 
quently a small percentage of lead is introduced to facilitate 
machining. Brass having the composition just stated may be 
drawn or rolled in the cold state. The cold drawing or rolling 
changes the structure of the metal, increasing its strength and 
brittleness, and consequently the original ductility must be 
restored by an annealing operation. 

48. Bronze. — Bronze is an alloy of copper and tin. Zinc is 
sometimes added to cheapen the alloy, or to< change its color and 
to increase its malleability. 



40 BRONZE [Chap. II 

(a) Commercial bronze. — Commercial bronze is acid-resisting 
and contains 90 per cent, of copper and 10 per cent, of tin. This 
metal has been used successfully for pump bodies, also for thrust 
collars subjected to fairly high pressures. Another bronze which 
has proven very serviceable for gears and worm wheels where 
noiseless operation is desired, contains 89 per cent, of copper and 
11 per cent, of tin. A form of bronze known as gun metal 
has the following approximate composition : 88 per cent, of copper; 
10 per cent, of tin; and 2 per cent, of zinc. It is used for high- 
grade bearings subjected to high pressures and high speeds. 

(6) Phosphor bronze. — Phosphor bronze varies somewhat in 
composition, but in general is about as follows: 80 per cent, 
copper; 10 per cent, tin; 9 per cent, lead; and 1 per cent, phos- 
phorus. It is easily cast and is as strong or stronger in tension 
than cast iron. It is a very serviceable bearing metal and is 
used for bearings subjected to heavy pressures and high speeds; 
for example, locomotive cross-head bearings, crankpin bearings, 
and bearings on grinders and blowers. 

A phosphor bronze intended for rolling into sheets or drawing 
into wire contains about 96 per cent, of copper, 4 per cent, of tin, 
and sufficient phosphorus to deoxidize the mixture. The tensile 
strength of such a phosphor bronze is equal to that of steel. 

(c) Manganese bronze. — By the term manganese bronze, as 
commonly used, is meant an alloy consisting largely of copper and 
zinc with small percentages of other elements such as aluminum, 
tin and iron. In reality many of the so-called manganese bronzes 
are not bronzes at all, but brasses; however, there are several 
compositions in use in which the proportion of zinc is small 
compared to the amount of tin and these are, strictly speaking, 
bronzes. Many of the commercial manganese bronzes contain 
no manganese whatever, the latter being used merely as a de- 
oxidizing agent. 

Due to its high tensile strength and ductility, manganese 
bronze is well adapted for castings where great strength and 
toughness are required. The hubs and blades of propellers and 
certain castings used in automobile construction are frequently 
made of this alloy. It is not nearly as satisfactory as phosphor 
bronze when used for bearings. A manganese bronze made of 
56 per cent, of copper, 43.5 per cent, zinc and 0.5 per cent, 
aluminum possesses high tensile strength and is suitable for the 



Art. 49] ALUMINUM 41 

service just mentioned. Manganese bronze may also be rolled 
into sheets or bars, or drawn into wire. 

(d) Aluminum bronze. — Aluminum bronze is formed by adding 
not to exceed 11 per cent, of aluminum to copper, thus producing 
an alloy having great strength and toughness. An alloy con- 
taining 90 per cent, of copper and 10 per cent, aluminum with a 
trace of titanium has given very satisfactory service when used 
for machine parts requiring strength and toughness, and at the 
same time subject to wear; for example, a worm wheel. The 
last-named composition produces an alloy that has an ultimate 
tensile strength equal to that of a medium-carbon steel. Ac- 
cording to tests made at Cornell University, the coefficient of 
friction of this type of aluminum bronze is 0.0018, thus making 
it suitable for bearings, and experience has shown that for accu- 
rately fitted bearings, the results are very satisfactory. The 
titanium in the above composition is added to insure good solid 
castings. In addition to the uses mentioned above, this type 
of bronze, due to its ability to resist corrosion, may be used for 
parts exposed to the action of salt water, tanning and sulphite 
liquids. 

49. Monel Metal. — Monel metal is a combination of approxi- 
mately 28 per cent, of copper, 67 per cent, of nickel and small per- 
centages of manganese and iron. It has a high tensile strength, 
is ductile, and has the ability to resist corrosion. It may be used 
to produce castings having an ultimate strength of 65,000 pounds 
per square inch. When used for rolling into sheets or bars, the 
strength is increased from 25 to 40 per cent. 

Monel metal presents no difficulties in machining, nor in 
forging operations if worked quickly. Like copper, it is impos- 
sible to weld it under the hammer, but it can be welded by means 
of the oxy-acetylene flame or by electricity. Since this alloy is 
non-corrodible it is used largely for propeller blades, pump rods, 
high-pressure valves, and steam-turbine blading. 

50. Aluminum. — Within the last few years aluminum alloys 
have been used rather extensively for many different machine 
parts. Pure aluminum is very ductile and may be rolled into 
very thin plates or drawn into fine wire. It may also be cast, 
but the casting produced has a coarse texture and for that reason 
pure aluminum is used but little for castings. For good com- 
mercial casting, aluminum alloys are used. The alloys recom- 
mended by the Society of Automobile Engineers are the following : 



42 BABBITT METAL [Chap. II 

(a) Aluminum copper. — The aluminum copper alloy contains 
not less than 90 per cent, of aluminum, 7 to 8 per cent, of copper, 
and the impurities consisting of carbon, iron, silicon, manganese 
and zinc shall not exceed 1.7 per cent. This is a very light 
material; is tough, possesses a high degree of strength, and may 
be used for castings subjected to moderate shocks. 

(b) Aluminum-copper-zinc. — An alloy, made of not less than 
80 per cent, of aluminum, from 2 to 3 per cent, of copper, not 
more than 15 per cent, of zinc, and not to exceed 0.40 per cent, of 
manganese, gives a light-weight, close-grained material that can 
be cast easily and will be free from blow holes. The castings pro- 
duced are very strong and are capable of resisting moderate 
shocks. 

(c) Aluminum zinc. — The alloy containing 65 per cent, alu- 
minum and 35 per cent, zinc is intended for castings subjected to 
light loads. It is quite brittle and is used for footboards and 
other similar parts of an automobile. It is about the cheapest 
aluminum alloy that is now in use. 

Due to the excessive shrinkage to which all aluminum castings 
are subjected, great care must be exercised in their design. Thick 
sections should never join thin sections on account of cracks that 
are very likely to show up in the finished castings. In order to 
obtain the best results, all parts should be given as nearly a con- 
stant or uniform section as is practical, and strength combined 
with light weight may be obtained by proper ribbing. 

Cast aluminum is used successfully in the construction of the 
framework of automobile motors, thus saving in weight and aid- 
ing in cooling the motor. It is also used for the construction of 
gear cases, pistons and clutch parts. For machine tools such as 
planers, pulleys are frequently made of cast aluminum so as to 
decrease the inertia of the rotating parts. The bodies or frame- 
work of jigs are occasionally made of cast aluminum, thus mak- 
ing them easier to handle. 

51. Babbitt Metal. — The term babbitt metal generally refers to 
an alloy consisting of copper, tin and zinc or antimony and in 
which the tin content exceeds 50 per cent. 

(a) Genuine babbitt metal. — The alloy containing copper, tin, 
and antimony is usually called genuine babbitt metal. Accord- 
ing to the Society of Automobile Engineers, the follownig specifi- 
cations will produce a high grade of babbitt that should give ex- 
cellent results when used for such service as connecting-rod 



Art. 52] WHITE BRASS 43 

bearings, automobile motor bearings, or any other machine 
bearings subjected to similar service: copper, 7 per cent.; anti- 
mony, 9 per cent. ; tin, 84 per cent. There are a large number of 
commercial grades of babbitt metals, many of which have a 
high percentage of lead and consequently sell at a low price. 

(b) White brass. — The alloy called white brass is in reality a 
babbitt metal, since its tin content exceeds 50 per cent., as the 
following specifications adopted by the Society of Automobile 
Engineers show: 3 to 6 per cent, of copper; 28 to 30 per cent, of 
zinc ; and not less than 65 per cent, of tin. This alloy is recom- 
mended for use in automobile engine bearings and generous lubri- 
cation must be provided to get the best results. 

Another well-known alloy of this type is that known as Parsons 
white brass, containing 2.25 per cent, of copper, 64.7 per cent, of 
tin, 32.9 per cent, of zinc and 0.15 per cent, of lead. It gives 
excellent service in bearings subjected to heavy pressures such as 
are found in marine and stationary engine practice; also in con- 
nection with high-speed service such as prevails in saw-mill 
machinery, paper and pulp machinery and in electric generators. 

As a rule, white brass is hard and tough, and to get the best 
results it must be poured at a very high temperature, and should 
then be peened or hammered all over before machining the 
bearing. 

HEAT TREATMENTS 

52. Heat-treating Processes. — The term heat treatment is 
applied to all processes of heating and cooling steel through cer- 
tain temperature ranges in order to improve the structure, and 
at the same time produce certain definite and desired character- 
istics. The processes involved in heat treatments are as follows : 

(a) Annealing. — The object of annealing steel is to remove the 
internal stresses due to cooling as well as to produce a finer tex- 
ture in the material. In general, annealing reduces hardness and 
increases the tensile strength and elongation of the steel. 

(b) Hardening. — Steel is hardened so as to produce a good 
wearing surface or a good cutting edge. The effect of hardening 
is to raise the elastic limit and the ultimate strength of the steel 
and at the same time reduce its ductility. When the carbon con- 
tent of the steel is 0.5 per cent, or over, the metal becomes brittle 
due to the stresses induced by the sudden quenching. 

(c) Tempering. — The process of tempering consists of reheating 



44 HEAT TREATMENTS [Chap. II 

the hardened steel in order to restore some of the ductility and 
softness lost in the hardening process. This means that the elas- 
ticity and tensile strength are reduced below the values for the 
hardened steel, but are higher than those prevailing in the original 
material. 

(d) Case-hardening. — By the process of case-hardening, the 
outer shell or skin of a piece of steel is converted into a high- 
carbon steel while the material on the inside remains practically 
unchanged. The type of steel to which this process is applied 
generally has a carbon content of 0.10 to 0.20 per cent., and ordi- 
narily should not contain more than 0.25 per cent, manganese or 
the case produced becomes too brittle. Case-hardening may 
also be applied to nickel steel, chrome steel, chrome-nickel steel, 
or chrome-vanadium steel. 

To case-harden, the pieces are packed in carbonaceous material 
in special boxes that are air-tight. These boxes with their con- 
tents are then placed in a furnace in which the temperature is 
brought up to about 1,500°F., and maintained at that tempera- 
ture for a definite time so as to produce the desired result. The 
pieces, after receiving this first treatment, may be quenched and 
are ready for use. In order to get better results, the boxes and 
their contents are allowed to cool in the furnace or in the air 
to about 1,200°, and are then again subjected to a high tem- 
perature after which the contents are quenched. A few of the 
materials used for packing the steel in the boxes are as follows: 
crushed bone; charred leather; barium carbonate and charcoal; 
wood charcoal and bone charcoal. 

The above method requires considerable time, and very often 
it is desirable to produce quickly a case-hardening effect which 
need not penetrate the material very far. This may be accom- 
plished by the use of a mixture of powdered potassium cyanide 
and potassium ferro cyanide, or a mixture of potassium ferro- 
cyanide and potassium bichromate. In general, case-hardening 
is used when a machine part must have a very hard surface in 
order to resist wear or impact, and when the interior of the piece 
must be tough so as to resist fracture. 

53. S. A. E. Heat Treatments. — In January, 1912, the Society 
of Automobile Engineers adopted a series of so-called heat treat- 
ments which they recommend for use with the various types of 
steel employed in automobile construction. Each heat treat- 
ment is designated by a letter and at the present time seventeen 



Art. 53] HEAT TREATMENTS 45 

different treatments are included in the above-mentioned list. 
The specifications are complete as may be seen from the follow- 
ing, taken from the Report of the Iron and Steel Division of the 
Standards Committee of the above-mentioned society. 

Treatment A. — For screws, pins and other similar parts 
made from 0.15 to 0.25 per cent, carbon steel, and for which 
hardness is the only requirement, the simple form of case-harden- 
ing designated as Heat Treatment A, will answer very well. After 
the piece has been forged or machined, treat it as follows : 

1. Carbonize at a temperature between 1,600° and 1,750°F. 

2. Cool slowly or quench. 

3. Reheat to 1,450° to 1,500°F. and quench. 

Treatment C. — Steel containing from 0.25 to 0.35 per cent, 
carbon is used for axle forgings, driving shafts, and other struc- 
tural parts, and in order to get better service from this grade 
of steel, the parts after being forged or machined should be heat- 
treated, the simplest form of which is given by the following 
specifications : 

1. Heat to 1,475° to 1,525°F. 

2. Quench. 

3. Reheat to 600° to 1,200°F. and cool slowly. 

In the third operation, namely that of drawing, each piece must 
be treated individually; for example, if considerable toughness 
with no increase in strength is desired, the upper drawing tem- 
peratures must be used; while with parts that require increased 
strength and little toughness, the lower temperatures will 
answer. 

Treatment K. — Treatment K, specifications for which are 
given below, is applicable to parts made of nickel and nickel- 
chromium steels, in which extremely good structural qualities are 
desired : 

1. Heat to 1,500° to 1,550°F. 

2. Quench. 

3. Reheat to 1,300° to 1,400°F. 

4. Quench. 

5. Heat to 600° to 1,200°F. and cool slowly. 

In reality this is a double heat treatment, which produces a 
finer structure of the material than is possible with only one 
treatment. 

Treatment V. — Springs made from silicon-manganese steel 
are treated as follows : 



46 GALVANIZING [Chap. II 

1. Heat to 1,650° to 1,750°F. 

2. Quench. 

3. Reheat to a temperature between 600°. and 1,400°F., and 
cool slowly. 

PREVENTION OF CORROSION 

To prevent corrosion of iron and steel it is necessary to protect 
the surfaces by means of some form of coating which may be 
either of a non-metallic or metallic nature. In the non-metallic 
method, the parts are coated with a paint, enamel or varnish, the 
efficiency of which depends on its being more or less air-tight. 
This method is far from satisfactory due to the chemical changes 
causing the coating to peel off or to become porous. In the 
metallic method the parts are coated with some other metal, 
generally zinc, though sometimes copper or aluminum, is used. 
There are three distinct processes of putting a zinc coating on 
iron and steel, as follows : hot-galvanizing, electro-galvanizing, and 
shererdizing . 

54. Galvanizing. — (a) Hot-galvanizing consists in dipping the 
parts, which have been cleaned previously, into molten spelter 
having a temperature of from 700° to 900°F. To cause the 
spelter to adhere to the surfaces of the articles, a soldering flux 
(metallic chlorides) is used. The zinc deposited on the parts is 
not chemically pure, and the impurities increase with continued 
use of the molten spelter. Due to these impurities, the coating 
is more or less brittle and will crack easily. The thickness of the 
coating is far from uniform. The process just described is the 
oldest known for coating iron and steel with zinc. 

(b) Electro-galvanizing . — This process is also known as cold- 
galvanizing and consists in depositing zinc on the parts, previously 
cleaned, by means of electrolysis. By this method any size of 
article may be treated, and it is claimed that the deposit consists 
of chemically pure zinc. The thickness of the coating is more 
easily controlled by this process than by the one discussed in the 
previous paragraph. 

55. Shererdizing. — In the process known as shererdizing, the 
articles, after they are cleaned, are packed with zinc dust in an 
air-tight drum. To prevent the oxidation of the zinc by the air 
inside of the drum, a small amount of pulverized charcoal is 
mixed with the contents of the drum. The drum, after being 



Art. 55] SHERERDIZING 47 

sealed, is placed into a specially constructed oven in which it is 
brought up to a temperature approximately 200° below the melt- 
ing point of zinc. To get an even distribution of the heat and 
at the same time to produce an even coating on the articles, the 
drum is rotated continually. By means of this process it is 
possible to produce a homogeneous deposit of zinc, the thick- 
ness of which depends upon the length of time the articles are 
allowed to remain in the oven. 



CHAPTER III 

FASTENINGS 
RIVETS AND RIVETED JOINTS 

56. Rivets. — The most common method of uniting plates, as 
used in boilers, tanks and structural work, is by means of rivets. 
A rivet is a round bar consisting of an upset end called the head, 
and a long part called the shank. It is a permanent fastening, 
removable only by chipping off the head. Rivets should in 
general be placed at right angles to the forces tending to cause 
them to fail, and consequently the greatest stress induced in them 
is either that of shearing or of crushing. If rivets are to resist 
a tensile stress, a greater number should be used than when they 
are to resist a shearing or a crushing stress. 

Rivets are made of wrought iron, soft steel, and nickel steel. 
They are formed in suitable dies while hot from round bars cut 
to proper length. The shank is usually cylindrical for about one- 
half its length, the remaining portion tapering very slightly. 
In applying rivets, they are brought up to a red heat, placed in 
the holes of the plates to be connected, and a second head is 
formed either by hand or machine work. Generally speaking, 
machine riveting is better than hand work, as the hole in the 
plates is nearly always filled completely with the rivet body, 
while in hand work, the effect of the hammer blow does not 
appear to reach the interior of the rivet, and produce a move- 
ment of the metal into the rivet hole. 

57. Rivet Holes. — For the sake of economy rivet holes are 
usually punched. There are two serious objections to thus 
forming the holes. The metal around the holes is injured by the 
lateral flow of the metal under the punch; however, this objection 
may be obviated by punching smaller holes and then reaming 
them to size. Secondly, the spacing of the holes in the parts to 
be connected is not always accurate in the case of punching, so 
it becomes necessary to ream out the holes, in which case the 
rivets may not completely fill the holes thus enlarged, or to use 
a drift pin. The drift pin should be used only with a light- 
weight hammer. 

48 



Art. 58] 



FORMS OF RIVETS 



49 



The diameter of the rivet hole is about 3^6 mcn larger than 
that of the rivet. This rule is subject to some variation, depend- 
ing upon the class and character of the work. The clearance 
given the rivet allows for some inaccuracy in punching the plate 
and in addition permits driving the rivet when hot. Drilling 
the holes is the best method for perforating the plates. The late 
improvements in drilling machinery have made it possible to 
accomplish this work with almost the same economy as in punch- 
ing. The metal is not injured by the drilling of holes; indeed, 
there are tests which show an increase in the strength of the metal 
between the rivet holes. 

58. Forms of Rivets. — Rivets are made of a very tough and 
ductile quality of iron or steel. They are formed in dies from 




the round bar while hot, and in this condition are called rivet 
blanks. For convenience, the head which is formed during the 
process of driving is called the point, to distinguish it from the 
head that is formed in making the rivet blank. The amount of 
shank necessary to form the point depends upon the diameter 
of the rivet. Since the length of the rivet is measured under the 
head, the length required is equal to the length of shank neces- 
sary to form the point plus the grip or thickness of plates to be 
riveted together. The various forms of rivet points and their 
proportions, as used in riveted joints, are illustrated in Fig. 7. 
In addition to the proportions for the points, the figure also gives 
the length of shank required to form these points. The style 
of point shown in Fig. 7(a) is called the steeple point; that illus- 
trated by Fig. 7(6) is known as the button point, while the counter- 
sunk point is represented by Fig. 7(c). The lengths of rivets 



50 FORMS OF HEADS [Chap. Ill 

should always be taken in quarter-inch increments on account of 
stock sizes. Any length up to five or six inches, however, may 
be obtained, but the odd sizes will cost more than the standard 
sizes. 

59. Forms of Heads. — Rivets with many different forms of 
heads may be found in mechanical work, but the ones in general 
use in boiler work are only three, namely, cone head, button head 
and countersunk head. These are shown in Fig. 7(a), (o), and 
(c), respectively. The proportions advocated by different manu- 
facturers vary somewhat; those given in Fig. 7 are used by the 
Champion Rivet Company. The steeple point, Fig. 7(a), is one 
easily made by hand driving and is therefore much used. This 
form, however, is weak to resist tension and should not be used 
on important work. 

The cone head, Fig. 7(a) is one of great strength and is used a 
great deal in boiler work. It is not generally used as a form for 
the point on account of difficulty in driving. The button-head 
type, Fig. 7(6), is widely used for points and may be easily formed 
in hand driving by the aid of a snap. The countersunk point 
weakens the plate so much that it is used only when projecting 
heads would be objectionable, as under flanges of fittings. Its 
use is sometimes imperative for both heads and points, but it 
should be avoided whenever possible. The countersink in the 
plate should never exceed three-fourths of the thickness of the 
plate, and for that reason, the height of the rivet point is generally 
from 3^6 to ~% inch greater than the depth of the countersink. 
The point then projects by that amount, or if the plate is required 
to be perfectly smooth, the point is chipped off level with the 
surface. 

RIVETED CONNECTIONS 

There are three general groups of riveted connections or joints: 
the first of these includes all types of joints met with in the 
construction of tanks and pressure vessels; the second group, 
commonly called structural joints, includes those that are com- 
mon to cranes, structures, and machinery in general; the third 
group includes those joints used in the construction of the hulls 
of ships. It is evident, that in the first group, in addition to 
forming a rigid connection between two or more members, the 
joint must also be made secure against leakage. In the third 
group mentioned, strength, stiffness and durability are the im- 



Art. 60] 



TYPES OF JOINTS 



51 



portant points desired, as well as proof against leakage; however, 
due to the low pressures the question of leakage presents no 
serious difficulties. 

60. Types of Joints. — Generally speaking, the following ar- 
rangements used in connecting plates by means of rivets are 
equally well adapted to the three groups of connections mentioned 
in the preceding paragraph. 

(a) Lap joints. — By a lap joint is meant an arrangement 
which consists of overlapping plates held together by one or more 
rows of rivets. If one row of rivets is used as shown in Fig. 
8(a), the arrangement is called a single-riveted lap joint, and with 
two rows as represented in Fig. 9, it is called a double-riveted 





lap joint. In the latter form of joint, the rivets may be arranged 
in two ways, namely staggered as shown in Fig. 9(a), or the so- 
called chain riveted, illustrated in Fig. 9(b). 

It is apparent that a load producing a tensile stress in a lap 
joint tends to distort the joint so that the two connected plates 
are practically in the same plane, thus inducing a bending stress 
in the plate as well as tensile and shearing stresses in the rivet. 
This distortion is not quite so marked in double-riveted lap 
joints, due to the additional stiffness given by the greater width 
of the overlap. 

(b) Butt joints. — When plates butt against each other and are 
joined by overlapping plates or straps, the connection is called 
a butt joint. Such a joint may have one plate on the outside, or 
one on the outside and another on the inside, as shown in Fig. 10. 
As in lap joints, the rivets may be grouped in one or more rows 
on each side of the joint, and in either the chain or staggered 



52 



TYPES OF JOINTS 



[Chap. Ill 







mm 




(a) 



P •—! 



^^ 



(b) 




Fig. XO. 



Art. 61] FAILURE OF JOINTS 53 

riveted arrangement, illustrated by Figs. 10 and 12, respectively. 
Butt joints having two cover plates are not subjected to the ex- 
cessive distortion found in lap joints, though poor workmanship 
may cause a small bending stress in the plates and a tension on 
the rivet. 

61. Failure of Joints. — In arriving at the intensity of stress in 
any of the types of joints discussed in Art. 60, we shall assume 
that the unit stress is uniform over the area of the resisting sec- 
tion, which, of course, is not absolutely correct for joints subjected 
to bending nor for those containing two or more rows of rivets. 
Furthermore, in the following discussion no allowance will be 
made for the additional holding power of riveted joints due to the 
friction between the plates. American designers pay no atten- 
tion to this, as experiments made at the United States Arsenal at 
Watertown seem to indicate that the joints will slip a slight 
amount at loads considerably less than those due to the working 
pressures. According to experiments made by Bach, the fric- 
tional resistance of a riveted joint may be taken approximately 
equal to 15,000 pounds per square inch of rivet area. 

Experience has shown that riveted joints may give way in any 
one of the following ways: 

(a) Shearing of the rivet. — In all lap joints and butt joints with 
one strap, the rivets tend to fail along one section; while in butt 
joints with two straps, failure tends to take place along two sec- 
tions. Thus in Fig. 8(a), the tendency would be for the rivet to 
fail along the line where the plates come into contact, and after 
failure, the condition would be represented by Fig. 8(6). Such a 
rivet is said to be in single shear, and in case two sections resist 
the shearing action, the rivet is in double shear. If P represents 
the force transmitted by one rivet, and d the diameter of the 
rivet after driving, then 

P = ^^ (38) 

(b) Crushing of the plate or the rivet. — If the rivet be strong 
enough to resist the shearing force, the plate or the rivet itself 
may fail by crushing, as shown at A in Fig. 11. The force upon 
the rivet is distributed over a semi-cylindrical area causing a dis- 
tribution of pressure upon this area about which very little is 
known. In the design of riveted joints it is customary to con- 
sider only the component of this pressure which is parallel to the 



54 



FAILURE OF JOINTS 



[Chap. Ill 



force upon the rivet, and to assume that it is distributed over the 
projected area of the rivet. 

The unit stress indicated by this crushing action is called a 
bearing stress, and representing it by Sb, it is evident that the 
force transmitted by one rivet is 



P = dtS, 



(39) 



► 



e 



e 



<& 



t± 



in which t represents the thickness of the plate. From (39), it 
follows that for any particular size of rivet and load P, the bear- 
ing stress depends upon the 
thickness of the plate; hence it 
is possible to have different 
bearing stresses in one joint 
when two or more plates of 
different thicknesses are con- 
nected together. 

(c) Tearing of the plate. — In 
a riveted joint subjected to a 
tension, the plates may be pulled 
apart along the line of rivets as 
shown at B in Fig. 11. Evidently 
the least area of the plate resist- 
ing this tension is the net sec- 
tion between consecutive, rivets. 
If p represents the pitch of the 
rivets, then the force transmitted by each rivet is 

P = (p - d)tS t (40) 

(d) Failure of the margin. — By the term margin, also called 
lap, is meant the distance from the edge of the plate to the center 
of the line of rivets nearest the edge, as shown by the dimension 
a in Fig. 11. Failure of the margin may occur by shearing of the 
plate along the lines in front of the rivet as shown at C in Fig. 11. 
With actual joints in use, failure in this way is not likely to occur. 
It follows that the shearing resistance offered by the plate is 
2 atS, ; hence, the force each rivet is capable of transmitting is 



P = 2 atS. 



(41) 



The margin may also fail by tearing open as shown at D in 
Fig. 11. This failure no doubt is due to the fact that in a joint 
subjected to tension, the material in front of the rivet behaves 



Art. 62] BOILER JOINT 55 

very much like a beam loaded at the center, thus causing the 
plate to fail by breaking open on the tension side, usually near the 
center. The truth of the above statement has been borne out by 
numerous experiments. A rule used considerably by designers 
is to make the margin never less than one and one-half times the 
diameter of the rivet, and experience has proven that joints de- 
signed in this manner seldom fail due to a weak margin. The 
Association of the Master Steam Boiler Makers recommends that 
for boiler joints the margin be made twice the diameter of the 
rivet. This marginal distance has proven very satisfactory in 
that no trouble has been experienced in making such a joint 
steam-tight by caulking. 

62. Definitions. — In the investigation of the stresses in a 
riveted joint, it is convenient to take a definite length of the joint 
as the basis for our calculations. This length may or may not be 
equal to the pitch. In joints having two or more rows of rivets, 
the distance between the rows is commonly called the back pitch, 
and its magnitude is approximately 70 per cent, of the pitch. 
An examination of Figs. 10 and 12 shows that there are certain 
groups or arrangements of rivets which are repeated along the 
entire length of the joint, and for convenience such a group of 
rivets may be called a repeating group and the length occupied 
by it a unit length of a riveted joint. In the analysis of any type 
of riveted joint, the force transmitted by such a repeating group 
generally forms the basis of all calculations. Another term used 
to a considerable extent in connection with riveted joints is the 
so-called efficiency, by which is meant the ratio that the strength 
of a unit length of a joint bears to the same length of the solid 
plate. 

RIVETED JOINTS IN BOILER CONSTRUCTION 

63. Analysis of a Boiler Joint. — One of the objects desired when 
designing an efficient boiler joint is to make the joint equally 
strong against failure by shearing, bearing and tension ;. however, 
certain modifications are necessary for economic reasons and, as 
a result, the actual joint as finally constructed in the shop will 
have a slightly lower efficiency than the one having uniform 
strength. In order to illustrate the method that may be followed 
in designing a joint having its resistance to shearing, bearing and 
tension approximately the same, assume the double-riveted lap 
joint shown in Fig. 9. From this figure it is evident that the 



56 BOILER JOINT [Chap. Ill 

length of a repeating group is p, the pitch of the rivets. We 
shall assume that the two plates are of the same thickness t, 
and that the margin was made of sufficient length to insure 
against its failure. 

The resistance P due to the shearing of the rivets in a unit 
length of the joint is 

P = ^ (42) 

The resistance due to crushing of the plate and the rivets is 

P = 2 dtSi (43) 

The area resisting tension is (p — d)t, and multiplying this by 
the unit stress, St, the total resistance against tension is 

P = (p - d)tS t (44) 

The three equations just determined may now be solved simul- 
taneously if it is desired to make the joint of equal strength. 
Combining (42) and (43), we obtain 

d = — =- (45) 

Equating (42) and (44), the pitch becomes 

>-' + *st (46) 

Equating (43) and (44), it follows that 

P-d + ^jj? (*7) 

Basing the size of the rivet upon (45) would lead to odd diame- 
ters that are not obtainable, since the commercial sizes vary 
by K6~ m ch increments from % i ncn to' 1% inches in diameter. 
Hence, with the use of commercial sizes of rivets, it is impossible 
to make the joint equally strong against the three methods of 
failure discussed above. Furthermore as the thickness of the 
plate increases, the diameter d calculated by (45) becomes' ex- 
cessively large, thus introducing serious difficulties in driving such 
a rivet. Having decided upon the size of rivet, the pitch may be 
determined by means of (46) and (47), but it may be necessary 
to modify the calculated pitch so as to insure a steam-tight joint. 
From this discussion it is apparent that the group of theoretical 
formulas derived above serves merely as a guide. 



Art. 64] 



EFFICIENCY OF BOILER JOINTS 



57 



In general, the method of procedure to be used in designing 
riveted joints is as follows: 

(a) Determine expressions for the various methods of failure. 

(6) Select a commercial size of rivet, so that it may be driven 
readily. 

(c) Having selected the size of rivet, determine whether the 
rivet will fail by shearing or by crushing. 

(d) Determine the pitch by equating the expression for the 
tearing of the plate to that giving the rivet failure. 

(e) Determine the probable efficiency of the joint. 

64. Efficiency of the Joint. — The efficiency of a riveted joint is 
defined as the ratio that the strength of a unit length of a joint 
bears to the same length of the solid plate. In the analysis of 
the double-riveted lap joint, it developed that there were three 
distinct ways that the joint could fail; hence, the efficiency of 
that joint depends upon the expression that gives the minimum 
value of P. In a double-riveted butt and double-strap joint, 
there are six ways that failure may occur and whichever is the 
weakest determines the probable efficiency of the joint. 

The strength of the solid plate of thickness t and unit length 
L is t LS t ] hence, the general expression for the efficiency of a 
riveted joint becomes 

E = ™° m? (48) 

tLbt 

The range of values for the efficiency E for the various types of 

joints used in boiler design is given in Table 9. These values 

may serve as a guide 

i ■ , • Table 9. — Efficiency of Boiler Joints 

in making assumptions 

that are necessary 

when designing joints 

for a particular duty. 

In case the actual or 

calculated efficiency 

does not agree closely 

with the assumed 

value, the joint will 

have to be redesigned, 

until a fair agreement 

is obtained. 

65. Allowable Stresses. — In order to design joints that will 
give satisfactory service in actual use, considerable attention must 







Efficiency 


Type of joint 


Min. 


Max. 


Lap joint. 


Single-riveted 

Double-riveted 

Triple-riveted 


45 
60 
65 


60 

75 

84 


Butt joint 

with two 

cover plates 


Single-riveted 
Double-riveted 
Triple-riveted 
Quadruple-riveted 


55 
70 
75 

85 


65 
80 
88 
95 



58 



ALLOWABLE STRESSES 



[Chap. Ill 



be given to the selection of the proper working stresses for the 
materials used. At the annual meeting of the American Society 
of Mechanical Engineers held in December, 1914, a committee 
appointed by that society presented an extensive report in which 
the question of the selection of the material is discussed very 
fully. The recommendations are as follows: 



Table 10. — Ultimate Sheaking 
Stresses in Rivets 


Table 11. — Thickness op 

Shell and Dome Plates 
after Flanging 


Kind of 


Ultimate shearing 


Diameter of shell 


Minimum 


rivet 


Single 
shear 


Double 

shear 






36 and under . . . 

36 to 54 

54 to 72 

72 and over .... 


H 

% 


Iron 

Steel 


38,000 
44,000 


76,000 
88,000 











(a) In the calculations for steel plates when the actual tensile 
strength is not stamped on the plates, it shall be assumed as 
55,000 pounds per square inch. 

(b) The ultimate crushing strength of steel plate shall be taken 
at 95,000 pounds per square inch. 

(c) In rivet calculations, the ultimate shearing strengths given 

in Table 10, and based on the 



Table 12. — Thickness op 
Joint Cover Plates 



Butt 



Thickness of 
shell plates 



Thickness of 
cover plates 



cross-sectional area of the 
rivet after driving, shall be 
used. 

(d) To obtain the allowable 
working stresses, the ultimate 
strengths given above must be 
divided by the so-called factor 
of safety, the value of which 
should never be less than five. 

66. Minimum Plate Thick- 
ness. — According to recom- 
mendations made by the Boiler 
Code Committee of the American Society of Mechanical Engi- 
neers, no boiler plate subjected to pressure should be made less 
than }4 inch thick, and the thicknesses given in Table 11 for 
various shell diameters may serve as a guide in designing work. 



yi to 1 Vs2 inclusive 

% and i% 2 

Ke and ^ 2 

M to Y\ 6 inclusive. . 

Y% and Y± 

% 

1 andlM 

IK 



Art. 67] 



SIZE OF RIVET HOLES 



59 



For the thicknesses of the cover plate for butt joints, the recom- 
mendations of this committee are given in Table 12. 





Table 13. — Recommended Size of Rivet Holes 






Diameter of rivet holes 


Plate 


Lap joint 


Double-strap butt joint 


thickness 


Single- 
riveted 


Double- 
riveted 


Triple- 
riveted 


Double- 
riveted 


Triple- 
riveted 


Quadruple- 
riveted 


M 


%\ 


% 


He 


%2 


% 


% 


% 


He 


A 


x Ae 


H 


% 


% 


A 'Vie 


% 


' l He 


x Y\e 


13 Ae 


A 


13 As2 


15 Ae 




Vie 


1 


15 Ae 


Vs 


7 A 


15 Ae 




15 A2 


iHe 


15 Ae 




y% 


1 


15 Ae 


lHe 




l Vs2 










/I 6 


He 












% 










lHe 


x He 










We 


K 










x He 










lHe 


Vs 










15 Ae 










We 


We 


1 










We 



In Table 13 are given the diameters of rivet holes for dif- 
ferent plate thicknesses and various types of joints, as determined 
from a study of actual joints used in the construction of boilers 
and pressure tanks. 

67. Design of a Boiler Joint. — It is required to design a triple- 
riveted double-strap butt joint for the longitudinal seam of a 
boiler 66 inches in diameter, assuming the working pressure as 



60 DESIGN OF A BOILER JOINT • [Chap. Ill 

150 pounds per square inch, and the ultimate tensile strength of 
the plates as 60,000 pounds per square inch. For the factor of 
safety, and shearing and crushing stresses, use the values recom- 
mended by the American Society of Mechanical Engineers. 

(a) The first step in the solution of this problem is to assume 
the probable efficiency of the joint, which according to Table 9 
may be taken as 85 per cent. 

(b) Determine next the thickness of the shell plates making 
proper allowances for the decrease in the strength of the shell 
due to the joint. The formula for the plate thickness is de- 
termined by considering the boiler a cylinder with thin walls 
subjected to an internal pressure, whence 

, _ PD 150 X 66 _ nA o.- v 

1 " 2ES t " 2 X 0.85 X 12,000 " U ' 4 * 5 moh - 

Selecting the nearest commercial size, the thickness of the shell 
plates will be made }4 inch. 

(c) The cover plates or straps of a triple-riveted butt joint for 
a 3^-inch shell should be J-{q inch thick, according to Table 12, 
and the diameter of the rivet hole as given in Table 13 will be 
1^6 inch, thus calling for 1-inch rivets. 

((d) According to the recommendations of the American Society 
of Mechanical Engineers, the following ultimate stresses will be 
used;$ s = 44,000 and Sb = 95,000, from which the following unit 
values are obtained: 

—^ = 39,000 pounds. 

dt'Si = % X He X 95,000 = 44,150 pounds. 
dtS b = 17 Aq X V 2 X 95,000 = 50,470 pounds. 

(e) Having arrived at the proper plate thicknesses and the 
diameter of the rivets, the resistances to failure of the joint must 
be investigated in order to establish the probable pitch of the 
rivets. A triple-riveted double-strap butt joint similar to that 
shown in Fig. 12 may fail in any one of the following ways: 

1. Tearing of the plate between the rivet holes in the outer 
row. — Using the notation prevailing in preceding articles, the 
magnitude of the resistance to failure by tearing of the plate 
between the rivet holes is 

P = (p- d)tS t (49) 



Art. 67] 



DESIGN OF A BOILER JOINT 



Gl 



2. Tearing of the plate between the rivet holes in the second 
row, combined with the failure of the rivet in the outer row. — 
An inspection of Fig. 12 shows that before the plate could fail 
between the rivets in the second row, the rivet in the outer row 
would have to fail either by shearing or by crushing, hence for 
this case two separate resistances are obtained as follows: 

4 



(P 



2d)tS t + 
2d)tS t + dt'S b 



S, 



(50) 

(51) 




Fig. 12. 



3. Shearing of all the rivets. — It is evident that in the triple- 
riveted butt joint shown in Fig. 12, four rivets are in double shear 
and one in single shear; hence, the magnitude of the resistance 
to failure is 



9 7rd 2 



(52) 



62 DESIGN OF A BOILER JOINT [Chap. Ill 

4. Crushing of all the rivets. — There are five rivets resisting 
crushing; hence, the expression for the resistance to crushing is 

P = (4 1 + t')dS b (53) 

5. Combined crushing and shearing. — The joint may also fail 
by the crushing of the four rivets on the inner and second rows, 
and the shearing of the rivet in the outer row; hence, the com- 
bined resistances of these rivets is 

P = 4 dtS b + ^- 8 (54) 

A joint of the type discussed above should be designed so that 
the strength of the critical sections increases as these sections 
approach the center of the joint. This condition is fulfilled by 
making the values of P obtained from (50) and (51) greater than 
that obtained by the use of (49) ; that is 

(p. - 2 d)tS t + ^- s >(p - d)tS t (55) 

(p - 2d)tS t + dt'S h > (p - d)tS t (56) 

From (55) it follows that the diameter of the rivet hole becomes 

d > ^f* (57) 

and simplifying (56), we find that 

t' > "f* (58) 

The expressions given by (57) and (58) must be satisfied, if it 
is desired to make the triple-riveted butt joint shown in Fig. 12 
stronger along the inner rows than at the outer rows. Having 
satisfied these equations by choosing proper values for d, t and 
t', the pitch p is determined by equating the minimum value of 
P, obtained by evaluating (52), (53) and (54), to that obtained 
from (49), and solving for p. 

Applying the principles just established to the data given 
above, we find that according to (57), the minimum value of d 
is 0.87 inch, and from (58) the minimum value of t' is 0.32 inch; 
hence it is evident that the values assumed above will insure 
increased strength of the joint along the inner rows. 

An inspection of the above formulas indicates that (54) gives 



Art. 68] RIVET SPACING 63 

the minimum value of P, and, after substitution, we find that 
P = 240,880 pounds. Inserting this value in (49) and de- 
termining the magnitude of the pitch, we get p = 9.09 inches, 
say 9 inches. 

The strength of the solid plate is 9 X Y A X 60,000 = 270,000 
pounds; hence, from (48), the efficiency 

940 880 

e = y^o = °- 892 ° r 89 - 2 p er cent - 

RIVETED JOINTS FOR STRUCTURAL WORK 

The design of riveted joints for structural work generally calls 
for the selection of the economical size of the members required 
to transmit the given force, in addition to the determination of 
the proper size and number of rivets to be used. In structural 
joints the size of the rivet depends in a general way upon the size 
of the connected members, but the usual sizes are %, Y± and % 
inch in diameter. Rivets larger than % inch cannot be driven 
tight by hand and since in structural work many of the joints 
must be put together in the field by hand riveting, it is evident 
that J4 mcn is the limiting size for this class of work. Tables 
giving the maximum size of rivets that can be used with the 
various sizes of structural shapes may be found in the hand books 
published by the several steel companies. 

68. Rivet Spacing.— In the spacing of rivets the following 
points must be considered: 

(a) If rivets are spaced too closely, the material between 
consecutive rivets may be injured permanently. 

(6) Too close spacing might interfere with the proper use of the 
snap or set during the driving operation. 

(c) Rivets that are spaced far apart prevent intimate contact 
between the members; water and dirt may collect and the joint 
may thus deteriorate by rusting. 

(d) Rivets are usually spaced according to rules dictated by 
successful practice, as the following will indicate. The minimum 
pitch between rivets is approximately three times the diameter 
of the rivet, and the maximum is given as sixteen times the 
thickness of the thinnest plate used in the joint. 

(e) For gauge lines used in connection with the various struc- 
tural shapes, the steel companies hand books should be consulted. 



64 STRUCTURAL JOINTS [Chap. Ill 

69. Types of Joints. — In general, it may be said that the 
various lap and butt joints used in structural work are very 
similar to those discussed in Art. 60. In addition to lap and butt 
joints, there are a great variety of riveted joints in which the 
several forms of structural shapes are joined together, either 
with or without the use of connecting plates commonly called 
gusset plates. Several common forms of such joints will be 
discussed. 

The following order of calculations is common to practically 
all structural joints: 

(a) From the magnitude of the load to be transmitted, de- 
termine the size of the member. 

(b) In general the diameter of the rivets to be used in the 
connection depends -upon the size of the connected members. 

(c) Determine the number of rivets required in each member 
to transmit the load in that member. This number depends 
upon the shearing and bearing stresses, whichever determines 
the method of failure. 

(d) The rivets in the joint must be arranged or spaced in such a 
manner that in the case of a tension member the stress along 
a section through a rivet does not exceed the allowable stress. 
To determine the net area in such a case it is customary to con- 
sider the size of the rivet hole to be % inch larger than the 
diameter of the rivet. For compression members, the area of 
the rivet hole is never considered in determining the net area of 
the member. 

70. Single Angle and Plate. — A very common method of con- 
necting a single angle, either in tension or compression, to a plate 
is shown in Fig. 13(a). It is apparent that the connection of one 
leg of the angle to the gusset plate will cause the angle to be 
loaded eccentrically; this eccentricity increases the stress con- 
siderably over that due to central loading. The determination 
of the additional stress due to the moment does not complicate 
the problem to any great extent, and for that reason the analysis 
necessary to determine the size of the angle in any given case 
should be made as complete as possible. The following problem 
will serve to illustrate the method of procedure in any given case. 

It is desired to determine the size of an angle and the number 
and size of rivets required in a connection similar to that repre- 
sented in Fig. 13, in which the force P acting on the member e 
is 16,800 pounds. Assume the allowable stresses in tension, 



Art. 70] 



ANGLE AND PLATE 



65 



shearing and bearing as 16,000, 10,000, and 20,000 pounds per 
square inch, respectively and the thickness of the gusset plate 
as Y± inch. 

The net area of the cross-section of the required angle, assuming 

central loading, must be 1ft ' nr)r> = 1-05 sq. in. This condition 

would be met by a 3 by 2j^> by ^-inch angle having a net area 
of 1.09 square inches after making allowance for a ^-inch rivet. 
Taking account of the eccentric loading, we will try a 3}i by 3 by 




p3'U 


r Xi 




r t 

6T4 



(b) 



Fig. 13. 



16,430 pounds per square inch 



%-mch angle, having a gross area of 2.30 square inches and a net 
area of 1.97 square inches. From a table of properties of struc- 
tural angles, we find that the distance Xi in Fig. 13(6) is 0.83 inch, 
thus making the eccentricity of the load P equal to 0.955 inch. 
Hence applying (17) the maximum tensile stress in the angle is 
16,800 16,800 X 0.95 X 0.-83 

1.97 + 1.85 

which is assumed as sufficiently close to the allowable stress given 
above. 

To obtain the number of rivets in the joint, determine whether 
the rivet is stronger in shear or in bearing. For the case con- 
sidered, the bearing resistance is the smaller, having a value of 
3,750 pounds per rivet; hence five ^-inch rivets are required. 

From the above analysis, it is apparent that the stress due to 



66 BEAM CONNECTIONS [Chap. Ill 

the eccentricity of the load P cannot be disregarded, and further 
that economy of material is obtained by loading the angle cen- 
trally. The latter condition is considered fulfilled when both legs 
are connected to the gusset plate. Such a connection is effected by 
the use of a clip angle g as shown in Fig. 13(c), provided the rivets 
are divided equally. Assuming that the joint is made similar to 
that shown in Fig. 13(c), the data given in the above problem 
calls for a 3 by 2^ by 34-inch angle. Each angle must be con- 
nected to the gusset plate by means of three rivets, and the same 
number must be used for connecting together the two angles. 
Tests made on steel angles having a clip-angle connection with 
the gusset plate, as illustrated in Fig. 13(c), do not confirm the 
analysis just given, since the results seem to indicate that very 
little is gained by the use of such angles. 

71. End Connections for Beams. — The-rivets in the connec- 
tions used on the ends of beams are subjected to a secondary 
shearing stress in addition to the direct stress due to the load on 
the joint, as the following analysis will show: 

According to the steel manufacturer's handbook the standard 
connection for a 12 by 40-pound I-beam consists of two 6 by 4 
by %-inch angles 73^ inches long, as shown in Fig. 14(a). Fur- 
thermore, the same source of information gives 8.2 feet as the 
minimum length of span for which the connection is considered 
safe when used with a beam loaded uniformly to its full capacity. 
The uniform load that the beam will carry without exceeding 
a fiber stress of 16,000 pounds per square inch is 

w 8 X 16,000 X 41.0 _ Q OQn , 

= 8.2 X 12 = ' pounds 

This gives a reaction R at the end connection of 26,665 pounds, as 
shown in Fig. 14(a). It is evident from an inspection of the 
figure that this reaction tends to rotate the connecting angles 
about the center of gravity of the rivet group, thus causing each 
rivet to be subjected to a shear due to the turning moment, in 
addition to the direct shear caused by the reaction. 

Due to the reaction R, the direct shear coming upon each 

rivet in the group has a magnitude of — '-= — or 5,333 pounds. 

Due to the turning moment, the shearing stress produced in any 
rivet in the group is proportional to the distance that the rivet is 
from the center of gravity of the group; hence, the resisting mo- 



Art. 71J 



BEAM CONNECTIONS 



67 



merit of each rivet about the center of rotation varies as the 
square of this distance. Letting $ s represent the secondary 
shear in the rivet nearest to the center of gravity, and l lf h, etc., 
the distances from the center of gravity G to the rivets 1, 2, etc., 
respectively, as shown in Fig. 14(6), then the external moment 
M, being equal to the summation of the resisting moments due 
to the rivets, is given by the following expression : 



M 



= -f[n+n+n+%+ii\ 



(59) 



e'WxIJs-TJiV 



(a) 




<T<k 



<) 



12x40 lb I Beam 




Fig. 14. 



From Fig. 14(6), the values h, U, etc., may be calculated, and 
since M is known, the magnitude of S' s is readily obtained. For 
the data at hand S'„ = 3,490 pounds; hence, the shears coming 
upon the various rivets are as follows: 

Secondary shear on rivet 1 = 3,490 lb. 

Secondary shear on rivet 2 = 10,300 lb. 

Secondary shear on rivet 3 = 7,140 lb. 

Secondary shear on rivet 4 = 7,140 lb. 

Secondary shear on rivet 5 = 10,300 lb. 

To determine the resultant shear upon each rivet, the direct 
and secondary shears must be combined. This may be done by 



68 DOUBLE ANGLE AND PLATE [Chap. Ill 

algebraic resolution, or graphically as shown in Fig. 14(c). It 
is evident that rivets 2 and 5 are subjected to the heaviest stress, 
the magnitude of which scaled from Fig. 14(c) is 13,150 pounds; 
whence the unit shearing stress in each of these ^-inch rivets is 
14,880 pounds per square inch. Since the web thickness of the 
12-inch by 40-pound beam is 0.56 inch, the bearing stress coming 
upon rivets 2 and 5 is 31,300 pounds per square inch. This 
problem shows the importance of determining the actual stresses 
in the rivets of eccentrically riveted connections. 

In the later editions of the steel manufacturer's hand books, it 
is of interest to note that the "End Connections for Beams and 
Channels" have been redesigned and for the size of beam given 
in the preceding problem two 4 by 4 by % 6-inch angles 8}4 
inches long are now recommended instead of those mentioned 
above, and furthermore only three %-inch rivets are used. 

72. Double Angle and Plate. — A form of connection met with 
occasionally is shown in Fig. 15. It is desired to determine the 
load P that this form of connection will safely carry, assuming that 
all rivets are %-inch in diameter and that the following stresses 
shall not be exceeded: S t = 15,000; S s = 10,000; S b = 20,000. 

The connection may fail in the following ways: 

(a) The rivets in the outstanding leg of the lug and girder 
angles may fail due to tension. 

(b) The rivets may shear off or crush in the vertical legs of the 
lug angle. 

(c) The rivets may shear off or crush in the angles A. 

(d) The lug angles may fail by combined tension and bending. 
The specifications for structural steel work do not recognize 

the ability of rivets to resist tension; however, for secondary 
members it is not unusual to assume the permissible stress in 
rivets subjected to tension as equivalent to the permissible shear- 
ing stress. Upon this assumption, the eight rivets in the out- 
standing legs of the lug angles are capable of supporting safely 
a load of 8 X 0.442 X 10,000 = 35,360 pounds. From the de- 
tails shown in Fig. 15, it is apparent that the rivets in the ver- 
tical legs of the lug angles and those in the angles A are of equal 
strength, hence the safe load that they are capable of support- 
ing, as measured by their resistance to crushing, is 3 X M X 
% X 20,000 or 16,875 pounds. 

To determine the bending stress in the lug angles it is assumed 
that the outstanding legs of these angles are equivalent to canti- 






Art. 72] 



DOUBLE ANGLE AND PLATE 



69 



levers having the load applied at the center of the rivets. Upon 
this assumption, the maximum bending moment occurs in the 
vertical leg, and its magnitude in this case is determined as 

P 

follows: Let ^y be the vertical load coming upon each inch of 

length of the lug angle; then since this load is considered as 
applied at the center of the rivet, the magnitude of the bending 
moment M per inch of length of the angle is 

M = 0.75^ (60) 

Equating this moment to the moment of resistance per inch 




2-5x3 X-J5UH2 lg . 



Fig. 15. 

of length, we obtain the following relation between the bending 
stress S't and M: 

18P 



S" 



(61) 



In addition to this flexural stress there is a direct stress S' t , the 
magnitude of which is 

S[ = ~ (62) 

The summation of the stresses given by (61) and (62), accord- 
ing to the conditions of the problem should not exceed 16,000; 
therefore 

„ 16,000 L 



19 



(63) 



70 SPLICE JOINT [Chap. Ill 

Since L = 12 inches, the maximum safe load that the angle will 
stand is, according to (63), equal to 10,100 pounds. 

Comparing this load with those determined for the other 
methods of failure, it is evident that the 10,100 pounds is the 
maximum load that can be supported safely by the connection 
represented in Fig. 15. 

73. Splice Joint. — In Fig. 16 is shown a form of joint used in 
the bottom chord of a Fink roof truss. Four members are joined 
together by means of a vertical gusset plate e and a splice plate / 
underneath the outstanding legs of the bottom chord angles. 
Due to the fact that a Fink truss is generally shipped in four 




pieces, the splice joint is made in the field. In the joint shown in 
Fig. 16, the magnitude of the loads upon the members a, b, c, 
and d are 30,100, 11,700, 13,000 and 17,700 pounds respectively; 
it is required to design the complete connection assuming the 
same working stresses as used in the problem of Art. 72, and 
furthermore, that no plate shall have a thickness less than y± inch. 
(a) Size of members. — In Table 14 are given the steps that 
are necessary in arriving at the sizes of the tension members 
a, c and d. Attention is called to the fact that the sizes of the 
members a and c are established by the loads given in Table 14 
and not by those given above. This is because certain mem- 
bers of light trusses are made continuous. According to certain 
specifications, the minimum size of angles used is 2 by 2 by Ya. 
inch while according to others, the minimum is 2^ by 2 by 34 
inch. In the present case the latter size is adopted, as this choice 
permits the use of %-inch rivets through the 23^-inch leg. 



Aet. 73] 



SPLICE JOINT 



71 



Table 14. — Tension Members 





Max. load 


Allow. 

stress 


Required, 
area 


Section selected 


Truss 
mem- 


No. 


Size 


Area 


ber 


Gross 


Net 


a 
c 
d 


36,600 
19,600 
17,700 


16,000 


2.29 
1.22 
1.11 


2 
2 
2 


3^X2^XM 
2MX2 XH 
2%X2 XH 


2.88 
2.14 
2.14 


2.45 
1.70 
1.70 



The size of the compression member b is arrived at in a general 
way by determining the allowable unit compressive stress by 
means of (25), having assumed a probable cross-section for the 
member in question. The area of the assumed section is then 
compared with that obtained by dividing the load on the member 
by the calculated unit stress. If the former area is equal to or 
slightly greater than the latter, the section assumed is safe. In 
determining the area of a compression member no reduction is 
made for the rivet hole, as it is assumed that the rivet in filling 
up the hole does not weaken the section. 

The allowable unit compressive stress is given by the following 
expression derived directly from (25) : 



16,000 



70 -> 

r 



(64) 



in which I denotes the length of the member in inches and r the 
least radius of gyration in inches. Generally the length of the 
compression members in roof trusses should not exceed 125 times 
the least radius of gyration. If, as in a roof-truss problem, it is 
required to determine the size of a series of compression members, 
the best method of procedure is to arrange the calculations in 
tabular form. In the above problem, the length of the member 
b is 93.8 inches, and the thickness of all plates will be assumed as 
34 inch. 

Assume the member b to be made of minimum size angles, 
namely, two 2}4 by 2 by 34 inch having an area of 2.14 square 
inches. The least radius of gyration r is 0.75 inch when the 
angles are arranged back to back with a 34-inch plate between 
them. This gives a ratio of I to r as 125 which is safe. The 
allowable working stress calculated by means of (64) is 7,250 

pounds per square inch; hence, the required area is = '„ or 

1.62 square inches. Since the area of the members chosen is in 



72 SPLICE JOINT [Chap. Ill 

excess of the calculated area, our assumption is on the side of 
safety. 

(b) Number of rivets. — The number of rivets required to 
fasten each of the members b and c to the gusset plate e is de- 
termined as explained in Art. 70, while the number required in 
the members a and d depends upon various assumptions that may 
be made. Among these are the following: 

1. The sum of the horizontal components of the forces in the 
members b and c, which is equal to the difference between the 
forces acting on the members a and d, is transmitted through 
the gusset plate e to the member a; hence, the number of rivets 
required to fasten a to the gusset plate is based on this force. 
It follows that the splice plate / and the rivets contained therein 
must be designed to transmit the total force in d. The vertical 
legs of the member d must also be riveted to the plate, but 
these rivets are not considered as a part of the splice. 

2. Consider that all of the rivets in the connection are effective, 
that is, the total number of rivets required in each of the members 
a and d must be based on the load transmitted by these members. 
This is equivalent to making the gusset plate transmit a certain 
part, say approximately one-half, of the load in d, and the re- 
mainder is taken up by the splice plate. Due to the fact that the 
splice plate is riveted to a and d by an even number of rivets, it 
frequently happens that the loads taken up by the splice and 
gusset plates are far from being equal. The method of pro- 
cedure is shown by the following problem: 

The size of the members will permit the use of ^-inch rivets 
throughout, except in the splice plate, where %-inch rivets must 
be used. We shall assume that four %-inch rivets are used at 
each end of the splice plate, and these are capable of transmitting 
4 X 2,045 or 8,180 pounds, or 46 per cent, of the load in the 
member d. If six %-inch rivets are used, the splice plate will 
then transmit 69 per cent, of the load in d. The former combina- 
tion is the one selected, as by its use the entire joint can be made 
up with fewer rivets than would be required if the second scheme 
were used. Now the remaining load in d, or 9,520 pounds, is 
transmitted through the gusset plate. The load in the member 
a minus the load transmitted by the splice plate is 21,920 pounds; 
this load must be transmitted through the gusset plate and re- 

21 920 
quires „ yr n or 6 shop rivets. The number of field rivets in 

the vertical legs of the member d is ' »» or 4. 



Art, .74] 



BOILER BRACE 



73 



The member b requires ^ '„ or 3 shop rivets while the member 
c needs n.L^ or 4 shop rivets. 

74. Pin Plates. — Not infrequently in structural work forces 
are transmitted from one member to another by means of pins, 
and in such cases the bearing area between the pin and members 
must be sufficient to transmit the load safely. A common case 
is that of channels through the webs of which passes a pin. In 
order to prevent the crushing of the webs, reinforcing or pin plates 
must be riveted to them. In arriving at the thickness of such 
pin plates, it is assumed that the load is distributed uniformly 
over the total bearing area, and that each plate is capable of tak- 
ing a load equal to the total load multiplied by the ratio that the 
thickness of the plate bears to the total thickness. Knowing the 
load coming upon each plate, the number of rivets required to 
fasten it to the web of the channel is readily obtained. Another 
example of the use of pin plates is shown in the reinforcing of 
the side plates of crane blocks. 

75. Diagonal Boiler Brace.— In Fig. 17 is shown a form of 
boiler brace used for connecting the unsupported area of the head 




Fig. 17. 



to the cylindrical shell. It consists of a round rod having 
flanged or flattened ends by means of which the brace is riveted 
to the head and shell. Due to the action of the steam pressure, 
the brace may fail in any one of the following ways: (1) The body 
of the brace may fail by tension ; (2) the flanged ends at the head 
may fail due to flexure, while the forged end at the shell may fail 
due to combined bending and direct tensile stresses; (3) the rivets 
may fail at the head end; (4) the rivets may fail at the shell end. 



74 BOILER BRACE [Chap. Ill 

(a) Failure of the brace body. — Letting P represent the force 

exerted upon the brace due to the pressure on the area supported 

by the brace, then the component of this force along the rod is 

P seca. Hence the stress in the rod is given by the following 

expression : 

4Pseca . . 

S t = —^- (65) 

(6) Failure of the brace ends. — 1. Head End. — The end at- 
tached to the boiler head may fail by bending of the outstanding 
legs. If 2 e represents the distance between the two rivets as 
shown in the figure, then the stress in the sections adjacent to 
the rod is 

S * = T^ (2e ~ a) (66) 

As usually constructed the type of brace shown in Fig. 17 is con- 
siderably stronger at the flanged ends than in the body. 

2. Shell end. — At the shell end it is customary to investigate 
the brace merely for direct tension. Representing the width 
of the flanged end by g and its thickness by /, then the tensile 
stress is 

.*'-*£?> (67) 

(c) Failure of the rivets at the head end. — The rivets at the 
head end of the brace are subjected to direct tensile, shearing, 
and bending stresses, the latter two of which are generally not 
considered in actual calculations. The force causing the tensile 
stress in the rivets is the total force P minus the area (I X c) 
multiplied by the steam pressure. However, since the shearing 
and bending stresses are not considered, it is customary to take 
the total force P as coming upon the two rivets. Hence the 
tensile stress in the rivets is 

Si = ^ 2 (68) 

The shearing stress coming upon the rivets is 

, 2 P tana , , 

S s = — gs- (69) 

If it is desired to find the resultant stress due to the combined 
effect of the two stresses just discussed, use the equations given 
in Art. 17. 



Art. 75] REFERENCES 75 

(d) Failure of the rivets at the shell end. — Due to the pull of 
the brace, the rivets at the shell end are subjected to shearing, 
tensile, and bending stresses. The first of these stresses is gener- 
ally the only one considered, since in the majority of cases the 
direct tensile and bending stresses are small. The component of 
the force in the rod at right angles to the rivets has a magnitude 
of P; hence, the shearing stress in the rivets, assuming that two 
rivets are used to fasten the brace to the shell, is 

S. = f| (70) 

(e) Allowable stresses. — The allowable shearing stresses in the 
rivets vary from 5,000 to 8,000 pounds per square inch, while the 
permissible tensile stresses in the diagonal brace proper vary from 
6,000 to 10,000 pounds per square inch. For the rivets in tension, 
the allowable stress should not exceed that given for shearing. 

References 

Design of Steam Boilers and Pressure Vessels, by Haven and Swett. 

Elements of Machine Design, by Kimball and Barr. I 

Elements of Machine Design, by W. C. Unwin. 

Die Maschinen Elemente, by C. Bach. 

Mechanics of Materials, by M. Merriman. 

Steam Boilers, by Peabody and Miller. 

Structural Engineer's Handbook, by M. C. Ketchum. 



CHAPTER IV 

FASTENINGS 
BOLTS, NUTS, AND SCREWS 

76. Forms of Threads. — The threads of screws are made in a 
variety of forms depending upon the use to which the screws are 
to be put. In general, a screw intended for fastening two or 
more pieces together is fitted with a thread having an angular 
form, while one intended for the transmission of power will have 
the threads either square or of a modified angular form. 

Two common forms of threads used for screw fastenings are 
the well-known V and the Sellers or United States Standard 
threads, shown in Fig. 18(a) and (b), respectively. Both forms 
are strong and may be produced very cheaply. Furthermore, due 




to their low efficiency, they are well adapted for screw fastenings. 
The proportions of these threads are given in the figures, and, as 
shown, the angle used is 60 degrees. The symbol p denotes the 
pitch, by which is meant the axial distance from a point on one 
thread to the corresponding point on the next thread ; or in other 
words, the pitch is the distance that the nut advances along the 
axis of the screw for each revolution of the nut. Evidently, the 
number of threads per inch of length is equal to the reciprocal of 
the pitch for a single-threaded screw. 

76 



Art. 76] FORMS OF THREADS 77 

(a) Sellers standard. — The form of thread shown in Fig. 18(6) 
is recognized as the standard in the United States, though the 
sharp V form is still in use. Due to the flattening of the tops 
and bottoms of the Vs in the Sellers standard, this form is much 
stronger than the sharp V thread. In Table 15 are given the 
proportions of the various sizes of bolts and nuts up to 3 inches 
in diameter, based on the Sellers standard. The Sellers system 
with some modifications has been adopted by the United States 
Navy Department. Instead of using different proportions for 
finished and unfinished bolt heads and nuts, the Navy Depart- 
ment adopted as their standard those given for rough work, thus 
permitting the same wrench to be used for both classes of bolts. 
In addition to this change, the Navy Department has adopted a 
pitch of 34 inch for all sizes above 2% inches, which does not agree 
with the Sellers system. 

(6) Standard pipe thread. — In Fig. 18(c) is shown a section of a 
standard pipe thread which may also be considered a form of fas- 
tening, though not for the same class of service as those discussed 
above. It will be noticed that the total length of the thread 
is made up of three parts. The first part designated as A in Fig. 

18(c) has a full thread over a tapered length of — '- ~,in 

which D represents the outside diameter of the pipe and n the 
number of threads per inch. The second part B has two threads 
that are full at the root but imperfect at the top and not on a 
taper. The part C includes four imperfect threads. The total 
taper of the threads is % inch per foot, or the taper designated 
by the symbol E is 1 in 32. It should be remembered that gas 
pipe goes only by inside measurement, that is, by the nominal 
diameter. The actual inside diameter varies somewhat from the 
nominal, but only the latter is used in speaking of commercial 
pipe sizes. 

(c) Square thread. — Three forms of screw threads that are 
well adapted to the transmission of power are shown in Fig. 19. 
The square thread shown in Fig. 19(a) is probably the most com- 
mon, and its efficiency is considerably higher than that obtained 
by the use of V threads. It has serious disadvantages in that it 
is very difficult to take up any wear that may occur, and further- 
more, it costs considerably more to manufacture. The pro- 
portions of square threads have never been standardized, but the 



78 



TABLE OF BOLTS AND NUTS [Chap. IV 





3 

a 
■d 
a 

as 

03 

l! 
'I 




'3 © 




« X X £ & X x ^ h § s « » £ & : x * s x s & £ " x 

r-lr-<>-lt-(.-li-(i-<r-l<N(N<N<N 




SB 

O as 


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HHHHHINNNININMMMn^Ttlirj 




la 

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t-\iH «\ « H IN O0\ COS cX t>K H f4\«!\ rf\H t-\ r4\ r\ H 00\ oX 

HrtHrtHrteiNflMIMMnM'*^ 


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P 

Q 

f5 


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13 

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1 


01 

a 


3 


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T3 


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a 

03 

■§ 

a 

Q 


C 
O 


03 
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02 


\00 \W \C0 *? \W CS. \t0 „ \S0 N \tO \tO « \t0 VO \tO ■* \P0 

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ci w so t-\i-Keo\^\o A h on w «\w w\ w »n\ o* to oo Ah 

HHHHHM««NMCO»Mti>Jt)liooffl 


[0 

Q 




^tD^N^ NMNOONJO «tO\CO V0»\tO\(O^\t0N#5SM 
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A H\ * "X * * H « <# t~\eO\»\r* C0\00 CT\N 00 C0\e0 C» rt 
SO rt 10 N HHHHHHNMMWNnMM<I^^IO 


1 
1 


o 
to 


SCO NrH S« \eo to to V* to to \iH 
\N Oi\ f^\ m\ \X r-f\ N^H V* NrH \» 09\ \r* \M S-H \-* >0\ \C0 \« \00 V^J VX) 
iH\ ,-t H « t-\ 00 J\ -H t-\ IO\ rt C0\ C0\ 0\ C0\ rf A A h\ A io\ 


J 


03 

oj O 


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ll 

5* 


i»'*o>'J , 0"30Nn«*!Oioco»OrtHON.cq« 

H«NtO*T|iil5CDt>t(l010HNMiiOh.9Ht|(<0 

ooooooooooOi-ir-ir-ii-ip-irHi-it-ie^NN 




■3 « 


fflHHHrtHHH 












V* SH \°0 \H C> Si- -» V* \00 s»0 V* \00 \CJ \00 \-* \00 V# \N V* 
i-K IC\ «\ t-\ -\ OX W\ C0\ t-\ -^ -\ <0\ r^\ W5\ «\ C-\ W\ rH\ S0\ 



Art. 76] 



ACME THREAD 



79 



practice of Wm. Sellers and Co., exhibited in Table 16, may 
serve as a guide. 





Table 


16. — Proportions op Sellers Square Threads 




Size 


Threads 


Root 


Size 


Threads 


Root 


Size 


Threads 


Root 


per inch 


diam. 


per inch 


diam. 


per inch 


diam. 


« 


10 


0.1625 


IK 


3M 


1.0 


3 


m 


2.5 


% 


8 


0.2658 


V6 


3 


1.2084 


3M 


m 


2.75 


y 2 


6M 


0.3656 


m 


2M 


1.4 


3M 


m 


2.962 


% 


5M 


0.466 


2 


2M 


1.612 


3% 


m 


3.168 


% 


5 


0.575 


2.14 


2K 


1.862 


4 


IX 


3.418 


% 


^A 


0.6806 


zy 2 


2 


2.0626 








i 


4 


0.7813 


m 


2 


2.3126 









(d) Trapezoidal thread. — The trapezoidal or buttressed thread 
shown in Fig. 19(&) is occasionally used for the transmission of 
power in one direction only. The driving face of the thread is at 



r-D -*£ 




right angles to the axis of the screw, while the back face makes an 
angle of 45 degrees, as shown in the figure. It is evident that the 
efficiency of this form of thread is the same as for a square thread, 
while its strength is practically that of the V thread. No stand- 
ard proportions have ever been devised or suggested, except those 
given in Fig. 19(6). 

(e) Acme thread. — The Acme thread is now recognized as the 
standard form of thread for lead screws and similar service, since 
the wear^can readily be compensated for by means of a nut split 



80 



MACHINE BOLTS 



[Ci 



IV 



lengthwise. Its efficiency is not quite as high as that of a square 
thread, but its cost of production is less since dies may be used in 
its manufacture. The form of the standard Acme thread is 
shown in Fig. 19(c), and in Table 17 are given the various dimen- 
sions indicated in the figure. 

Table 17. — Proportions of Acme Standard Threads 



Threads 
per 
inch 


a 


b 


t 


Threads 
per 
inch 


a 


b 


t 


10 


0.0319 


0.0371 


0.0600 


3 


0.1183 


0.1235 


0.1767 


9 


0.0361 


0.0413 


0.0655 


2V 2 


0.1431 


0.1483 


0.2100 


8 


0.0411 


0.0463 


0.0725 


2 


0.1801 


0.1853 


0.2600 


7 


0.0478 


0.0529 


0.0814 


m 


0.2419 


0.2471 


0.3433 


6 


0.0566 


0.0618 


0.0933 


IK 


0.2914 


0.2966 


0.4100 


5 


0.0689 


0.0741 


0.1100 


i 


0.3655 


0.3707 


0.5100 


4 


0.0875 


0.0927 


0.1350 


H 


0.7362 


0.7414 


1.0100 



SCREW FASTENINGS 

In general, screw fastenings are used for fastening together 
either permanently or otherwise two machine parts. To ac- 
complish this end, the following important forms are met with in 
machine construction; (a) bolts; (6) cap screws; (c) machine 
screws; (d) set screws; (e) studs; (f) patch bolts; (g) stay bolts. 

77. Bolts. — A bolt is a round bar one end of which is fitted with 
a thread and nut, while the other end is upset to form the head. 
Bolts are well adapted for fastening machine parts rigidly; but 
at the same time they allow the parts to be easily disconnected. 
Whenever conditions or surroundings will permit, bolts should 
be used for fastening machine parts together. 

(a) Machine bolts. — What is known as a machine bolt has a 
rough body, but the head and the nut may be rough or finished, 
as desired. Commercial forms of machine bolts are shown in 
Fig. 20(a) and (o). The heads and nuts may be square as shown 
in Fig. 20(a), or the hexagonal form shown in Fig. 20(6) may be 
used. The standard lengths of machine bolts as given in the 
manufacturers' catalogs are as follows : 

1. Between 1 and 5 inches the lengths vary by 3"i-inch 
increments. 

2. Between 5 and 12 inches, the lengths vary by 3^-inch 
increments. 









Art. 77] 



COUPLING BOLTS 



81 



3. Above 12 inches, the lengths vary by 1-inch increments. 

Any length of bolt, however, may be obtained, but odd lengths 
cost more than standard lengths. The length of the threaded 
part is from three to four times the height of the nut. 





r ^ 



& 



i m 


N 


w 


71 


i « 






(a) (b) 

Fig. 20. 



a 



(c) 



Table 18. — Coupling Bolts 



The proportions of the heads and nuts as used on standard 
machine bolts are given in Table 15. 

(6) Carriage bolts. — The carriage bolt, another form of through 
bolt, is shown in Fig. 20(c). 
Its chief use is in connec- 
tion with wood construc- 
tion. 

(c) Coupling bolts. — A 
coupling bolt is merely a 
machine bolt that has been 
finished all over, so that it 
may be fitted into reamed 
holes of the same diameter 
as the nominal diameter of 
the bolt. Coupling bolts 
are intended for use in 
connection with the best forms of construction. They are more 
expensive to produce and at the same time are more costly to 
fit into place. In Table 18 are given the dimensions of com- 
mercial sizes of such bolts. According jto the manufacturers' 







Head and nut 






Threads 






Size 


per 
inch 


Short 
diam. 


Thick- 
ness 


lengths 


V2 


13 


Vs 


V2 


2 -m 


% 


11 


IK 6 


% 


2 -5 


% 


10 


1% 


% 


2M-5M 


% 


9 


IKe 


% 


2M-5M 


1 


8 


1% 


1 


2%-5% 


1% 


7 


l 13 Ae 


IX 


3 -6 


W 


7 


2 


m 


3M-6 



82 



AUTOMOBILE BOLTS 



IChap. IV 



lists, coupling bolts are made with hexagonal heads and nuts 
only, and the lengths for the sizes listed in Table 18 run from 
2 inches up to 6 inches, varying by quarter-inch increments. 

(d) Automobile bolts. — The various bolts discussed in the 
preceding paragraphs have not been found satisfactory in auto- 
mobile construction, and in order to fulfill the requirements of 
strength and space limits demanded in this class of work, the 
Society of Automobile Engineers has adopted a special standard. 
The design of this type of bolt is shown in Fig. 21, and the data 




r— a — , — . k 



Fig. 21. 

in Table 19 give the various detail dimensions for the different 
sizes that have so far been standardized. It will be noticed that 
the heads and nuts are hexagonal, and that the thread is the 





Table 19 


— S. A 


. E. 


3tani 


)ARD 


Bolts 


AND 


Nuts 






Size 


Threads 
per 
inch 


Head of bolt 


Castellated nut 


Plain 
nut 


» 


6 


c 


d 


e 


/ 


9 


h 


Cotter 
pin 


k 


H 


28 
24 


Vie 

K 


He 
x Ve± 


%2 
Vei 


He 


Ve* 


%2 




Ha. 


Vie 


K? 


He 


2 Ve* 


%2 


*%4 


Vs 
Vie 


24 
20 
20 
18 
18 
16 
16 
14 
14 


He 
Vs 
H 
Vs 

15 Ae 
1 

Wie 
1Kb 


%2 

2 ^4 

H 

2 %4 

X %2 

3 %4 

He 

2 Vz2 

K 


Vs 


Vz2 


X %2 

2 %4 


K 


Vs 


H2 


2 ^4 


H 


He 

3 %4 


He 


Vie 


He 


%2 


Vs 


s Ha 


Vs 

l Vie 

H 

Vs 
l 


2 %2 
4 %4 

% 

2 %2 

1 


K 


3 VeA 
X %2 
2 V 3 2 
4 %A 


1H 
IK 


12 

12 


1% 


15 Ae 


%2 


Hi 


1H* 

IK 


He 


VZ2 


% 


6 %4 
1%2 




12 
12 


2 
2He 


IM2 


K 


He 


1% 


Vs 


K 


Wet 


1 13 ^4 

1Mb 



Sellers standard. Instead of using the same pitch as that recom- 



Art. 78] 



CAP SCREWS 



83 



mended in the Sellers system, a finer pitch has been adopted; 
furthermore, the heads and nuts are made somewhat smaller. 
The heads are slotted for a screw driver and the nuts are recessed 
or castellated so they may be locked to the bolt by means of 
cotter pins. 

78. Screws. — Screws, unlike bolts, do not require a nut, but 
screw directly into one of the pieces to be fastened, either the 
head or the point pressing against the other piece. The types of 
screws that hold the pieces together by the pressure exerted by 




~~.&f 



(a) 



(b) 



® 







IJ 



(e) 



(?) 



60 

m 



I© J L.J 



(c) 




CD 



(d) 




Fig. 22. 



the head of the screw are called cap screws and machine screws, 
while those whose points press against a piece and by friction 
prevent relative motion between the two parts are called set 
screws. By the term length of a screw is always meant the 
length under the head. 

(a) Cap screws. — Cap screws are made with square, hexagonal, 
round or filister, flat and button heads, and are threaded either 
United States Standard or with V threads. The various forms 
for heads are shown in Fig. 22, and in Table 20 are given the 



84 



CAP SCREWS 



[Chap. IV 



general dimensions of the commercial sizes that are usually kept 
in stock. All cap screws, except those with filister heads, are 
threaded three-fourths of the length for one inch in diameter or 
less and for lengths less than four inches. Beyond these dimen- 
sions, the threads are cut approximately one-half the length. 
The lengths of cap screws vary by quarter-inch increments 
between the limits given in Table 20. 

Cap screws, if properly fitted, make an excellent fastening for 
machine parts that do not require frequent removal. To insure 
a good fastening by means of cap screws, the depth of the tapped 
hole should never be made less than one and one-half times the 
diameter of the screw that goes into it. In cast iron, the depth 
should be made twice the diameter. 







Table 


20. — Standard Cap Screws 










Square head 


Hexagon head 


Socket head 




Threads 
per 














Size 






















inch 


Short 
diam. 


Thick- 


Length 


Short 
diam. 


Thick- 


Length 


Diam. 


Thick- 


Length 


H 


20 


Va 


A 


I 


He 


H 


} 


Va 


34 


M-334 


He 


18 


He 


He 


\K-a 


a 


He 


\K-3 


He 


He 


%-m 


Va 


16 


a 


H 


J 


He 


Va 


I 


He 


Va 


H-i 


He 


14 
13 


He 
% 


He 

a 


}«-4 


A 
A 


He 

A 


JM-4 


Ha 
A 


He 

a 


A-±A 


He 
% 


12 
11 


1Mb 

3 A 


He 

Va 


1 -4 

i -m 


13 Ae 

Va 


He 

Va 


1 -4 
1 -4K 


13 Ae 
Va 


He 

Ha 


}l -e 


3 A 


10 


A 


A 


m-m 


1 


A 


lA-^A 


1 


A 


1M-6 


Vs 


9 


iy* 


A 


1H-5 


ia 


Va 


1K-5 








1 


8 


IK 


1 


1H-5 


134 


1 


1M-5 








U4 


7 
7 




134 


V 2 -5 


m 

1H 


1H 


U -5 


















Round and 


filister 


E 


utton head 




Mat head 




Threads 
per 


head 












Size 






















inch 


Diam. 


Thick- 


Length 


Diam. 


Thick- 
ness 


Length 


Diam. 


Thick- 
ness 


Length 


H 


40 


He 


A 


S A~2A 


As2 


Ye 4 


A-1A 


34 




A-i 3 A 


He 


24 


A 


He 


K-2*i 


He 


A 2 


s A-2 


Va 




H-2 


A 


20 


% 


a 


$i-3 


He 


A 2 


A-2A 


i5 A2 




H-2H 


He 


18 


He 


He 


A-3A 


He 


%2 


A-2y 2 


Va 




H-2H 


% 


16 


He 


Va 


V±-1A 


Va 


He 


H.-2H 


H 




3 A-3 


He 


14 


Va 


He 


A-ZA 


*A 


Va 


K-3 


13 Ae 




1 -3 


a 


13 


A 


a 


Ji-4 


1 He 


Ws2 


1 -3 


Ha 




134-3 


He 


12 


13 Ae 


He 


1 -434 


1 Ae 


X %2 


134-3 


1 




1H-3 


% 


11 


A 


A 


1K-4W 


1 


a 


134-3 


m 




l%-3 


H 


10 


1 


A 


lM-4« 


m 


Va 


1K-3 


IVa 




2 -3 


Va 


9 


1H 


H 


l%-5 














1 


8 


m 


l 


2 -5 















Art. 78] 



MACHINE SCREWS 



85 



(b) Machine screws. — Machine screws are strictly speaking 
cap screws, but the term as commonly used includes various forms 
of small screws that are provided with a slotted head for a screw- 
driver. The sizes are designated by gauge numbers instead of 
by the diameter of the body. The usual forms of machine 
screws are shown in Fig. 23, 
and in Table 21 are given 
the dimensions of stock sizes. 

There are no accepted 
standards, each manufac- 
turer having his own. It I , \ , 1 /j\ 
should also be observed that 
machine screws have no 
standard number of threads, 
hence in dimensioning these 
screws, always give the num- ( a ) (t>) 
ber of the screw, the number Fig. 23. 
of threads and the length, 

thus No. 30 — 16 X 1M inches M. Sc. It may be noted that 
machine screws larger than No. 16 are not used extensively in 
machine construction; for larger diameters than No. 16, use 
cap screws. 



Sfe 



(c) 








f 




m 




j 


\ 


/ 









(c) (d) 

Fig. 24. 



(e> 



(f) 



The American Society of Mechanical Engineers has adopted 
a uniform system of standard dimensions for machine screws, 
but as yet they are not in universal use in this country. The 
report of the committee which was appointed to draw up such 
standards may be found on page 99 of volume 29 of the 
Transactions. 

(c) Set screws. — Set screws are made with square heads or 
with no heads at all, and may be obtained with either United 
States Standard or V threads. The short diameter of the square 



86 



TABLE OF MACHINE SCREWS 



[Chap. IV 





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0~S) 
<gg 




t-\ <m ~K CO ^\ 

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\CO M * VJO \0O V* vg, s^ » \=0 v* 




-a 

V 

5 


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53 


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ft 

Q 


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speaq Jaijsijg uo SB auiBS aqx 


03 
ID 


60 

h1 


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"JMOooomoaiiOHtDHiOHioHOHN 

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oddddddddddddddddo'd 


1 

s 


1631 
1894 
2158 
2421 
2684 
2947 
3210 
3474 
3737 
4263 
4790 
5316 
5842 
6368 
6895 
7421 
7421 
7948 
8474 


ooooooooooooooooooo 


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a 
o 


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53 


ft 

p 


COOO<NCOOlOO-*OOt><OTjH-*<N>-l©rooOCO 

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03 

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1544 
1786 
2028 
2270 
2512 
2754 
2996 
3238 
3480 
3922 
4364 
4806 
5248 
5690 
6106 
6522 
6938 
7354 
7770 


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COC0COCOC0-*rH'*-*"OiOCOCOI>t-l>0000O3 

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03 


3 
I 

Hi 


iOOC0<Nt~C000Tj<©r-l(NC0-*c0t>0005©lN 

M1005O01OOHN«T|im!DM10OlONM 

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s 


1350 
1561 
1772 
1984 
2195 
2406 
2617 
2828 
3040 
3462 
3884 
4307 
4729 
5152 
5574 
5996 
6419 
6841 
7264 


OOOOOOOOOOOOOOOOOOO 


•5"§ 

<* a 
J* 


64, 56, 48 
56,48 

} 40, 36, 32 

36, 32, 30 
32,30 
36, 32, 30 

} 32, 30, 24 

24,20 
24, 20, 18 

J20, 18, 16 

}l8, 16 
18, 16, 14 

|l6, 14 


a 




s 

03 
P 


NeoiOOMOrtW^OOHT|ihOT)(NOMO 
ilNOtOCDOMIOOlOfflOOTliHNMOfflCI 
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dddddddddddododddod 


1 


i-li-l^.-'rHNNNNNM 



Art. 78] 



SET SCREWS 



87 



heads as well as the height of the heads is made equal to the 
diameter of the body of the screw. The commercial lengths of 
set screws having heads vary from % to 5 inches by quarter inches. 
The headless set screws shown in Fig. 24(d) to (/) are made 
only in the following sizes: % by % inch; 3^ by %6 inch; % by 
1 K6 inch; and Y± by % inch. 

The principal distinguishing feature of set screws is the form of 
the point. The points are generally hardened. Only cup and 
round point set screws (see Fig. 24(a) and (&)) are regular, all 
other types being considered special. Set screws used as fasten- 
ings are not entirely satisfactory for heavy loads, and hence should 
only be used on the lighter loads. The cup point shown in Fig. 
24(a) has a disadvantage in that it raises a burr on the shaft thus 
making the removal of the piece, such as a pulley, more difficult. 
In place of the cup point, the conical point shown in Fig. 24(e) 
is frequently used, but this necessitates drilling a conical hole in 
the shaft, which later on may interfere with making certain de- 
sirable adjustments. 

To obtain the appropriate size of set screw for a given diameter 
of shaft, the following empirical formula based upon actual in- 
stallations may be found useful: 



diameter of set screw = - + ^{q inch, 

o 



(71)' 



in which d represents the diameter of the shaft. 

The question of the holding capacity of set screws has received 
little attention and about the only 
information available is that pub- 
lished by Mr. B. H. D. Pinkney' 
in the American Machinist of Oct. 
15, 1914. His results are based 
upon some experiments with Y^r 
and 3^2-inch set screws, in which 
he found that the latter size had a 
capacity of five times the former. 
With this fact as a basis, Mr. 
Pinkney calculated the data given 
in Table 22. Experience with the headless variety of set screws 
seems to indicate that due to the difficulty of screwing up, the 
holding power is somewhat less than for the cup and flat point 
type. 



Table 


22. — Safe Holding Ca- 


pacities op Set Screws 


Size 


Capacity, 




Capacity 




pounds 




pounds 


M 


100 


% 


840 


Vie 


168 


K 


1,280 


% 


256 


Vs 


1,830 


Vie 


366 


l 


2,500 


V 2 


. 500 


IX 


3,288 


He 


658 


IK 


4,198 



88 



PATCH BOLTS 



[CllAV: IV 





A 




r— B »-p 


t C 




'111 


III 


I \ 


lilt 


f 


111 


11 


SL 1 


ill 11 


1) 



(d) Studs. — A stud is a bolt in which the head is replaced by 
a threaded end, as shown in Fig. 25. It passes through one of 
the parts to be connected, and is screwed into the other part, 
thus remaining always in position when the parts are disconnected. 
With this construction the wear and crumbling of the threads 
in a weak material, such as cast iron, are avoided. Studs are 

usually employed to secure 
the heads of cylinders in en- 
gines and pumps. 

There is no standard for 
the length of the threaded 
ends of studs; hence, the 
length must always be speci- 
fied. Studs may be obtained finished at B or rough, and the 
ends threaded either with United States Standard or V threads. 
The commercial lengths carried in stock vary from \y± to 6 
inches by quarter inches for the finished studs. For the rough 
studs, the lengths vary from 3^ to 4 inches by quarter inches, and 
from 4 to 6 inches by half inches. Usually one end is made a 
tight fit, while the other is of standard size. 

(e) Patch bolts. — A form of screw commonly called a patch 
bolt is shown in Fig. 26(a); its function is that of fastening 
patches on the sheets of boilers. The application of a patch 



Fig. 25. 





(b) 



Fig. 26. 



bolt is illustrated in Fig. 26(6). Patch bolts should be used only 
when, due to the location of the patch, it is impossible to use 
rivets, as for example on the water leg of a locomotive boiler. 
As shown in Fig. 26(6), patch bolts are introduced from the side 
exposed to the fire and are screwed home securely. The head, 
by means of which they are screwed up, is generally twisted off 
in making the fastening. Instead of having the form and num- 
ber of threads according to the United States Standard, all stock 
sizes have 12 threads per inch of the sharp V type. 



Art. 79] 



STAY BOLTS 



89 



79. Stay Bolts. — -(a) Stay bolts are fastenings used chiefly in 
boiler construction. Due to the unequal expansion and con- 
traction of the two plates that are connected, stay bolts are 
subjected to a peculiar bending action in addition to a direct 
tension. As a result of the relative motion between the two 
connected plates, stay bolts develop small cracks near the inner 
edge of the sheets. These cracks eventually cause complete 
rupture, though it may not be noticed until the plates begin to 
bulge. Three types of stay bolts are shown in Fig. 27, the first 






(a) 



Fig. 27. 

of which is used extensively on small vertical and locomotive 
types of boilers. To provide some slight degree of flexibility 
and thereby decrease the danger of cracking near the plates, 
stay bolts are made as shown in Fig. 27(b). 

According to the Code of Practical Rules, covering the con- 
struction and maintenance of stationary boilers, recently adopted 
by the American Society of Mechanical Engineers, "each end of 
stay bolts must be drilled with a ^{Q-inch. hole to a depth extend- 
ing % inch beyond the inside of the plates, except on small vertical 
or locomotive-type boilers where the drilling of the stay bolts 
shall be optional." The object of these holes is to give some 
indication of a rupture by the leakage of the fluid. 

In Fig. 27(c) is shown one of the various types of so-called 
flexible stay bolts used in locomotive boilers. 



90 NUT LOCKS [Chap. IV 

(6) Stresses in stay bolts. — The area of the surface supported 
by a stay bolt depends principally upon the thickness of the 
plates and the fluid pressure upon the surface. Quoting from the 
Code of Rules adopted by the American Society of Mechanical 
Engineers, "the pitch allowed for stay bolts on a flat surface and 
on the furnace sheets of an internally fired boiler in which the 
external diameter of the furnace is over 38 inches, except a corru- 
gated furnace, or a furnace strengthened by an Adamson ring or 
equivalent," may be determined by the following formula, but 
in no case should it exceed 8}4 inches: 



in which 



p = > p+i)f + 6 , (72) 

C = constant having a value of 66. 

P = working pressure in pounds per square inch. 

t = thickness of plate in sixteenths of an inch. 

In addition to the formula just given, the above-mentioned 
Code of Rules contains tables and other formulas pertaining to 
the subject of staying surfaces that may be found useful in de- 
signing pressure vessels. 

Having determined the pitch of the stay bolts, a simple cal- 
culation will give the magnitude of the load coming upon each 
bolt. Dividing this load by the allowable stress, the result is 
the area at the root of the thread. For mild-steel or wrought- 
iron stay bolts up to and including 13^ inches in diameter, the 
American Society of Mechanical Engineers recommends that the 
allowable stress shall not exceed 6,500 pounds per square inch, 
and for larger diameters 7,000 pounds per square inch is recom- 
mended. The majority of screwed stay bolts have 12 threads 
per inch of the V type, though the United States Standard form 
is also used. 

80. Nut Locks. — Since nuts must have a small clearance 
in order to allow them to turn freely, they have a tendency to un- 
screw. This tendency is especially evident in the case of nuts 
subjected to vibration. In order to prevent unscrewing, a great 
many different devices have been originated, a few of which are 
shown in Figs. 28 to 30 inclusive. 

(a) Lock nut. — The cheapest and most common locking de- 
vice is the lock nut shown in Fig. 28. Two nuts are used, but 
it_is not necessary that both of these shall be of standard thick- 



Aet. 80] 



LOCK NUT 



91 



ness, as frequently the lower nut is made only one-half as thick 
as the upper one. Some engineers maintain that the lower nut 
should be standard thickness while the upper one could be thinner. 
The following analysis, due to Weisbach, shows that conditions 
might arise for which the first arrangement would answer, while 
for other conditions, the second arrangement would be the proper 
one to use. 

We shall assume that the lower nut in Fig. 28(a) has been 
screwed down tight against the cap c of some bearing. Denote 
the pressure created between the nut b and the cap c by the 
symbol P. Now screw down the upper nut a against b as tightly 
as the size of the stud or bolt d will permit, thus developing a 
pressure between the two nuts, which at the same time produces 
a tensile stress in that part of the stud d that comes within the 
limits of action of the two nuts. Designate the magnitude of 
this pressure between the nuts by the symbol P . Considering the 





Fig. 28. 



forces acting upon the nut b, it is evident that the force Po acts 
downward, while the force P acts upward, and the resultant 
force having a magnitude P — P acts upon the threads of the 
stud. Now the direction of this resultant depends upon the 
magnitudes of P and P . If P>Po the resultant force on the 
threads of the stud is upward, or in other words the upper 
surfaces of the threads in the nut b come into contact with the 
lower surfaces of the threads on the stud. From this it follows 
that when P>P , the nut b should be of standard thickness as 
shown in Fig. 28(a), since it alone must support the axial load. 
Let us consider the case when P Q >P; we shall find that the 
resultant force on the threads is downward, thus indicating that 



92 



COLLAR NUT 



[Chap. IV 



the lower surface of the threads in the nut b bear on the upper 
surfaces of the threads on the stud; hence, the upper nut must 
take the axial load and for that reason should be made of standard 
thickness as shown in Fig. 28(6). 

Now consider another case might arise, namely in which 
P Q = P. It is evident that the resultant is zero, thereby showing 
that no pressure exists on either the upper or lower surfaces of 
the thread; hence, the nut a carries the axial load P. 

On the spindles of heavy milling machines and other machine 
tools, the double lock nut is used to a great extent. The nuts are 
made circular rather than hexagonal and are fitted with radial 
slots or holes for the use of spanner or pin wrenches. 

(&) Collar nut. — The collar nut, shown in Fig. 29(a), has been 
used very successfully in heavy work. The lower part of the nut 





(b) 



Fig. 29. 



is turned cylindrical, and upon the surface a groove is cut. The 
cylindrical part of the nut fits into a collar or recess in the part 
connected. This collar is prevented from turning by a dowel 
pin as shown in the figure. A set screw fitted into the collar 
prevents relative motion between the latter and the nut. In 
connecting rods of engines, for example, where the bolt comes 
near the edge of the rod, the bolt hole is counterbored to receive 
the cylindrical part of the nut, and the set screw for locking the 
nut is fitted directly into the head of the rod. 

The following formulas have proved satisfactory in propor- 
tioning collar nuts similar to that shown in Fig. 29(a) : 



Art. 80] 



SPLIT NUT 



93 



inch 



2.25 d + . . 

1.5 d 

1.45 d 

0.75 d 

0.55 d 

0.2 d + K6 inch 

0.1 d + 0.1 inch 



(73) 



(c) Castellated nut. — Another effective way of locking nuts, 
used extensively in automobile construction, is shown in Fig. 
21. It is known as the castellated nut, and the commercial sizes 
correspond to the sizes of automobile bolts discussed in Art. 
77(d) . Attention is directed to the fact that, due to the necessity 
of turning the nut through 60 degrees between successive lock- 
ing positions, it may be impossible to obtain a tight and rigid 





connection without inducing a high initial stress in the bolt. 
The general proportions of the standard castellated nuts approved 
and recommended by the Society of Automobile Engineers are 
given in Table 19. 

(d) Split nut. — The double nut method of locking is not 
always found convenient due to restricted space, and in such 
places, the forms of nut locks shown in Fig. 30 have been found 
very satisfactory. In Fig. 30(6) is illustrated a hexagonal nut 
having a saw cut extending almost to the center. By means of a 
small flat-head machine screw fitted into one side of the nut, the 
slot may be closed in sufficiently to clamp the sides of the thread. 
The nut, instead of being hexagonal in form, may be made cir- 
cular, and should then be fitted with radial slots or holes for a 
spanner wrench. 

(e) Spring wire lock. — The spring wire lock shown in Fig. 30 (a) 
is another locking device adapted to a restricted space. This is 
a very popular nut lock for use with the various types of ball 



94 



WASHERS 



IChap. IV 



Table 23. — Plain Lock Washers 



bearings. The spring wire requires the drilling of a hole in the 
shaft, and in case any further adjustment is made after the nut 
is fitted in place, it requires drilling a new hole. A series of 
such holes will weaken the shaft materially. 

(/) Lock washer. — A nut lock used considerably on railway 
track work, and within recent years in automobile work, is shown 
in Fig. 29(6). It consists essentially of one complete turn of a 
helical spring placed between the nut and the piece to be fastened. 
When the nut is screwed down tightly, the washer is flattened out 
and its elasticity produces a pressure upon the nut, thereby pre- 
venting backing off. In Table 23 is given general information 

pertaining to the standard light 
and heavy lock washers adopted 
by the Society of Automobile 
Engineers. 

81. Washers. — The function of 
a washer is to provide a suitable 
bearing for a nut or bolt head. 
Washers should not be used unless 
the hole through which the bolt 
passes is very much oversize, or 
the nature of the material against 
which the nut or bolt head bears 
necessitates their use. For com- 
mon usage with machine parts, 
wrought-iron or steel-cut washers 
are the best. When the material 
against which the nut bears is rel- 
atively soft, such as wood for ex- 
ample, the bearing pressure due to the load carried by the bolt 
should be distributed over a considerable area. This is accom- 
plished by the use of large steel or cast-iron washers. 
^Washers are specified by the so-called nominal diameter, by 
which is meant the diameter of the bolt with which the washer 
is to be used. 

82. Efficiency of V Threads. — Before discussing the stresses 
induced in bolts and screws due to the external loads and to 
screwing up, it is necessary to establish an expression for the 
probable efficiency of screws. 





Section of washer 


Size of 


Light 


service 


Heavy 


service 


bolt 












Width 


Thick- 
ness 


Width 


Thick- 
ness 


He 






He 


%4 


K 






H ■ 


Vie 


He 






% 


Vie 


H 


Hi 


He 


Vz2 


%2 


He 


Vz2 


He 


%2 


%2 


y 2 


Vz2 


He 


%2 


%2 


He 






He 


%2 


% 


He 


Y%2 


Yi 


He 


3 A 


He 


y?.2 


Yt 


y± 


Vs 


H 


H 


K 


X 


1 


X 


Vb 


Yi 


K 



Art. 82] 



EFFICIENCY OF V THREADS 



95 



Let N = unit normal pressure. 

Q = axial thrust upon the screw. 
d = mean diameter of the screw. 
p = pitch of the thread. 
a = angle of rise of the mean helix. 
. j8 = angle that the side of the thread makes with the 
axis of the screw. 
// = coefficient of friction between the nut and screw. 
7] = efficiency. 

Consider a part of a V-threaded screw, as shown in Fig. 31, in 
which the section CDE is taken at right angles to the mean helix 
AO. The line OF represents the line of action of the normal 




Fig. 31. 

pressure N acting upon the thread at the point 0, and OF is 
drawn parallel to the axis of the screw. 

The vertical component of the normal pressure N acts down- 
ward and has a magnitude of iVcos y. The vertical component 
of the force of friction due to the normal pressure N acts upward, 
and its magnitude is ju'JV sin a. The algebraic sum of these two 
vertical forces gives the magnitude of the component of Q acting 
at the point 0. Thus 

Q = SiV (cos 7 — //sin a), 

rom which the total normal component is 

2N = Q-r-. — - (74) 

cos 7 — /x sin a 

In one revolution of the screw, the applied effort must be capa- 



96 EFFICIENCY OF V THREADS [Chap. IV 

ble of doing the useful work Qp and. overcoming the work of 
friction. Denoting the total work put in by the effort in one 
revolution of the screw by the symbol W t , we find that 

W, = Qv+ *» , *"T'. a (75) 

^ COS 7 — //sin a 
Substituting the value of p = 7rdtano:in (75), we get 

W t = irdQ Ttan a + M ' S6C a , . (76) 

L cos 7 — //sin a J 

Now to determine a relation between the angles a, (3 and 7, 
we make use of a theorem in Solid Analytical Geometry, namely, 

COS 2 7 + COS 2 (8 -f- COS 2 - — a = 1, 

from which 

cos 7 = \/cos 2 a — cos 2 /3 (77) 

Substituting (77) in (76), the following expression for the 
total work required per revolution of the screw, in order to raise 
the load Q, is obtained : 

W t = irdQ [tan a + . '*'«* £ . 1 ( 78 ) 

L V cos 2 a — cos 2 /3 — fi' sin a] 

By definition, the efficiency is the ratio of the useful work to 
the total work; hence, for the F-threaded screw 

tan a 



u sec a 

tan - 



(79) 



-\/cos 2 a — cos 2 j8 — //sin a 



Very often it is desirable to determine the magnitude of the 
effort P required at the end of a lever or wrench. Representing 
the length of the lever by L, and equating the work done by 
P in one revolution to the total work done, we find that 

P = ^*^i (80 ) 

2 Lit] 

STRESSES IN SCREW FASTENINGS 

To arrive at the proper dimensions of bolts, screws and studs 
used as fastenings, it is important to consider carefully the follow- 
ing stresses: 



Art. 83] STRESSES IN SCREWS 97 

(a) Initial stresses due to screwing up. 

(b) Stresses due to the external forces. 

(c) Stresses due to combined loads. 

83. Stresses Due to Screwing Up. — The stresses induced in 
bolts, screws and studs by screwing them up tightly are a tensile 
stress due to the elongation of the bolt, and a torsional stress due 
to the frictional resistance on the thread. To determine the mag- 
nitude of the resultant stress induced in a fastening subjected to 
these stresses, combine them according to Art. 17. For screws less 
than % inch in diameter, the stresses induced by screwing up 
depend so much upon the judgment of the mechanic that it is 
useless to attempt to calculate their magnitude. 

Experiments on screws and bolts have been made with the 
hope that the results obtained would furnish the designer some 
idea as to the magnitude of the stresses due to screwing up. As 
might be expected, the results varied within rather wide limits 
so that no specific conclusions could be drawn; however, all 
such tests seemed to show that the stresses are high, generally 
higher than those due to the external forces and very frequently 
running up to about one-half of the ultimate strength of the bolt. 

, 84. Stresses Due to the External Forces. — (a) Direct stress. — 
Bolts, screws, and studs, as commonly used for fastening machine 
parts, are subjected to a direct tensile stress by the external 
forces coming upon them; but occasionally the parts fastened 
will produce a shearing action upon the fastening. 

Assuming that a certain force Q causes a direct tensile stress 
in a bolt or screw, it is evident that the weakest section, namely 
that at the root of the thread, must be made of such a diameter 
that the stress induced will not exceed the allowable tensile 
stress. Calling the diameter at the root of the thread d , we 
obtain from (3) 

do = J^ (81) 

Table 15 gives the values of d for the various sizes of the 
Sellers standard threads. Since this table also gives the area at 
the root of the thread, the calculations for the size of a bolt for a 
given load Q is considerably simplified by finding the ratio of Q 
to St which is really the root area of the required size of both; 
then select from Table 15 the diameter corresponding to the 



98 



TRESSSES IN SCREWS 



[Chap. fV 



Screws subjected to a shearing stress should be avoided as far 
as possible. However, such an arrangement can be used success- 
fully by the use of dowel pins fitted accurately into place after 
the screws have been fitted. There are many places where 
dowel pins cannot be used, and for such cases it is suggested that 
the body of the bolt or screw be made an accurate fit in the 
holes of the parts to be fastened. 

Assuming as above that the external force coming upon the 
bolt is Q, and that the allowable shearing stress is S s , then it 
follows that 

(82) 

(6) Tension due to suddenly applied loads. — The loads pro- 
ducing the stresses discussed in the preceding paragraphs were 
considered as steady loads; however, bolts and screws are used 
in many places where the loads coming upon them are in the 



=31—1 iiii 

iffll Hit 1 1 tttnttl itti 

;^xJl W III 



(a) 



(b) 



Fig. 32. 



nature of shocks, as for example in the piston rod of a steam 
hammer, and in the bolts of engine connecting rods. Such bolts 
must then be designed so as to be capable of resisting the shocks 
due to the suddenly applied loads without taking a permanent 
set. Now since the energy of the suddenly applied load must be 
absorbed by the bolt, and as the measure of this energy is the 
product of the stress induced and the total elongation, it is evi- 
dent that the stress may be reduced by increasing the elongation. 
Increasing the elongation may be accomplished in several ways, 
among which are the following : 

1. Turn down the body of the bolt so that its cross-sectional 
area is equal to the area at the root of the thread; then since the 
total elongation of the bolt depends upon the length of this re- 
duced section it follows that the length of the latter should be 
made as great as possible. Such a bolt is weak in resisting tor- 
sion and flexure, and instead of fitting the hole throughout its 



Art. 85] STRESSES IN SCREWS 99 

length, it merely fits at the points where the body was not turned 
down, as shown in Fig. 32(a). Low cost of production is the 
chief advantage. 

The tie rods used in bridge and structural work are generally 
very long, and the prevailing practice calls for upset threaded 
ends, which is merely another way of making the cross-section of 
the body of the rod practically the same as the area at the root 
of the thread. No doubt in this class of work the object of 
making the rods as thus described is to save weight and mater- 
ial; however, it should be pointed out that the capacity for 
resisting shocks has also at the same time been increased. 

2. Instead of turning down the body, the cross-sectional area 
may be reduced by drilling a hole from the head of the bolt to- 
ward the threaded end, as shown in Fig. 32(&). This method 
no doubt is the best, as the bolt fits the hole throughout its length, 
and the hollow section is well-adapted to resist flexural as well 
as torsional stresses. The cost of production may be excessive 
for long bolts and for the latter the method of Fig. 32(a) may be 
employed. 

Actual tests were made by Prof. R. C. Carpenter at the Sibley 
College Laboratory on bolts 1% inches in diameter and 12 inches 
long, half of which were solid and the remainder had their 
bodies reduced in area by drilling a hole as shown in Fig. 32(6). 
Two of these bolts, tested to destruction, showed that the solid 
or undrilled bolt broke in the thread with a total elongation of 
0.25 inch. Additional tests in which similar bolts were subjected 
to shock gave similar results. 

85. Stresses due to Combined Loads. — Having discussed the 
individual stresses induced in bolts and screws by screwing them 
up and by the external loads coming upon them, it is in order 
next to determine the stress induced by the combined action of 
these loads. This resultant stress depends upon the rigidity of 
the parts fastened as well as upon the rigidity of the screw itself. 

(a) Flanged joint with gasket. — In general it may be said that 
for an unyielding or rigid bolt or screw fastening two machine 
parts that will yield due to screwing up, the stress in the bolt is 
that due to the sum of the initial tension due to screwing up and 
the external load, as the following analysis will show. In actual 
fastenings used for machine parts, neither the bolt nor the parts 
fastened fulfill the above conditions absolutely; however, the 
conditions are very nearly approached when some semi-elastic 



100 



STRESSES IN SCREWS 



[Chap. IV 



material like sheet packing is used to make a tight joint, as in 
steam and air piping. In a joint such as illustrated in Fig. 33(a) 
the packing acts like a spring, and tightening the nut will com- 
press the packing a small amount, thus causing a stress in the 
bolt corresponding to this compression. Assuming that an ex- 
ternal load due to some fluid pressure acts upon the flange a, its 
effect will be to elongate the stud thereby increasing the stress, 
and at the same time reduce slightly the pressure exerted upon 
the stud by the packing; hence, it follows that for this case, the 
load upon the stud may for all practical purposes be considered 
as equivalent to the sum of the two loads. 

(o) Flanged joint without gasket. — The next case to be con- 
sidered is that type of fastening in which the stud, bolt or screw 





(a) 



(b) 



Fig. 33. 



yields far more than the connected parts. This case is repre- 
sented by two flanges having a ground joint, as shown in Fig. 
33(6). Due to screwing up of this joint, the stud which now 
elongates, in other words acts like a spring, will be subjected to a 
stress corresponding to this elongation. If, as in the preceding 
case, we now introduce a pressure upon the flange b which tends 
to pull the fastening apart, it is evident that the resultant pres- 
sure at the ground joint is the difference between the pressures 
exerted by the nut c upon the outside of the flange b and that due 
to the fluid pressure on the inside of 6. As long as the pressure 
on the inside of b does not exceed that due to the screwing up of 
the nut, the stud will remain unchanged in length; hence the 
stress induced is that due to the initial tension and not that due 
to the external load. If, however, the pressure on the inside is 



Art. 86] 



STRESSES IN SCREWS 



101 



sufficient to overcome that due to the nut, the joint will separate, 
causing the stud to elongate; hence the stress in the latter is that 
due to the external load. 

86. Fastening with Eccentric Loading. — (a) Rectangular base. 
— In Fig. 34(a) is shown the column of a drill press bolted to 
the cast-iron base by cap screws. Due to the thrust P of the 
drill which tends to overturn the' column, these screws are sub- 
jected to a tensile stress which is not the same for each screw, 
as the analysis below will show. 

To determine the maximum load that may come upon any 
screw we shall assume that the column, which is rigid, is fastened 




U 



-1 42 



3 4 



3 4 

4 * 



Fig. 34. 



to the rigid base by means of eight screws, as shown in Fig. 34(6). 
Due to the thrust P, the column will tip backward about the 
point A, thus stretching each screw a small amount depending 
upon its distance from the axis AB, Fig. 34(6). Since the stresses 
induced in the screws vary directly as the elongations, it is 
evident that the loads upon the screws vary. 

Now the moment of the thrust P must be balanced by the sum 
of the moments of the screw loads about the axis AB; hence, 
representing the loads upon the screws by Qi, Q 2 , etc., and their 
moment arms by l\, h, etc., it follows that 



PL 



(83) 



2(QiZi + Q2I2 + Qzh + W) 

The subscripts used correspond to the number of the screw 
as shown in Fig. 34(6). Since the stresses induced in any screw 



102 



STRESSES IN SCREWS 



[Chap. IV 



vary directly as the elongation produced, we obtain the fol- 
lowing relations : 

Q* = Qi r Qs = Qi l r Q* = Qi l r (84) 

l\ L\ ii 

Substituting these values in (83), the expression for the exter- 
nal moment becomes: 

2Q 



PL -x 1 [fi + s + s+"fl] 



(85) 



From the preceding discussion, it is apparent that the maxi- 
mum stresses occur in the screws labeled 4, and the magnitude 
of this maximum stress is given by 
the following expression: 
PLh 




Qi = 



2di + % + n +® 



(86) 



Knowing the various dimensions, 
as well as the thrust P, the magnitude 
of Qi is readily determined, and from 
this the size of screw for any allow- 
able fiber stress. 

(b) Circular base. — Instead of hav- 
ing a rectangular base as discussed 
above, columns or machine members 
are frequently made with a circular 
base similar to that shown in Fig. 35, 
in which 2a represents the outside 
diameter of the column flange, and 
26 the diameter of the bolt circle. For the case under dis- 
cussion six bolts or screws numbered from 1 to 6 inclusive are 
used. Adopting a notation similar to that used in the preceding 
analysis, we have that the external moment due to the load P is 



Fig. 35. 



pl = I 1 (% + n + n + n .+ \\ + id 

From the geometry of the figure 

l\ = a — b cos a 

h = a — b cos (60 + a) 

h = a + b cos (60 — a) 

h = a + b cos a 

h = a + b cos (60 + a) 

U = a — b cos (60 — a) 



(87) 



(88) 



Art. 87] 



STRESSES IN SCREWS 



103 



Substituting these values in (87), it follows that 

6 a 2 + 3 6 2 



pl . Ql \^L+lHLl 

la — b cos ad 



from which the magnitude of Qi is given by the following expres- 
sion: 

*-»KiTiE! (89) 

Now to determine the maximum value of Qi for a given mo- 
ment PL and dimensions a and b, it is evident from (89) that this 
occurs when cos a is a minimum, i.e., cos a = — 1, which is the 
case whemthe angle a is 180 degrees. Hence 



_ PLv a + b I 
Wl " 3 l_2a 2 + &J 



(90) 



Knowing the maximum load, the size of the bolts or screws 
must be proportioned for this load. 




Fig. 36. 



By means of an analysis similar to the above, the stresses in 
any number of bolts or screws may be arrived at. 

87. Common Bearing. — In machinery, many forms of fasten- 
ings are used in which the bolts or screws are subjected to shear- 
ing stresses in addition to tensile stresses. A very simple form 
of such a fastening is shown in Fig. 36, which represents a solid 
cast-iron flanged bearing frequently found on heavy machine 
tools. Due to the power transmitted by the gears located on the 
shaft, the bearing is subjected to a pressure P which tends to 



104 POWER SCREWS [Chap. IV 

produce a shearing stress in each of the screws. For convenience, 
all of the screws are assumed to be stressed equally. As men- 
tioned in Art. 83, dowel pins may be used as shown in Fig. 36, 
and if these are fitted correctly they will, to a great extent if not 
altogether, relieve the screws from a shearing action. 

Due to the eccentric location of P, relative to the supporting 
frame, the bearing is subjected to an external moment PL, 
which must be balanced by an equal moment due to the tension 
set up in the screws. For the bearing shown in Fig. 36 having six 
screws on a bolt circle of diameter 2b, the relation between the 
external moment and the moment of the screw loads may be 
obtained from (90). 

Now assume a diameter of screw, and determine the direct 
shearing stress, if no dowel pins are used, also the tensile stress 
caused by the external moment. To arrive at the maximum in- 
tensity of stress, combine the two separate stresses by means of 
(28); the result should not exceed the assumed safe working 
stress. 

POWER SCREWS 

Three forms of threads adapted to the transmission of power 
are shown in Art. 76; of these the square thread is looked upon 
with the greatest favor due to its higher efficiency. Instead of 
having single-threaded screws, it is not unusual to employ screws 
having multiple threads, an example of which is shown in the 
friction spindle press illustrated in Fig. 125. In connection with 
multiple-threaded screws, attention is called to the terms lead 
and divided pitch. By the former is meant the distance that the 
nut advances for one revolution of the screw, and by the latter, 
the distance between consecutive threads ; hence a triple-threaded 
screw of one and one-half inch lead has a divided pitch of one- 
half inch. 

88. Efficiency of Square Threads. — Referring to Fig. 37, let d 
represent the mean diameter of the screw. The action of the 
thread upon the nut is very similar to the action of a flat pivot 
upon its bearing, and hence we shall assume that the pressure 
between the screw and the nut may be considered as concen- 
trated at the mean circumference of the thread. 

(a) Direct motion. — Representing the average intensity of 
pressure between the screw and its nut by the symbol q, we get 
for the total pressure on a small area 8A of the surface of the 



Art. 



EFFICIENCY OF SQUARE THREADS 



105 



thread q8A. If the screw is rotated so that the axial load 
Q is raised, as for example in a screw jack, the pressure qbA will 
act along the line OB making an angle <p' with the normal OA. 
The symbol <p' represents the angle of friction for the surfaces in 
contact. Now since the normal OA is inclined to the axis of the 
screw by the angle a, the angle of rise of the mean helix, it is 
evident that the components of the pressure qdA parallel to the 
axis and at right angles thereto, are as follows : 




Fig. 37. 



Hence 
or 



Parallel component = qbA cos (a + <p') 
Right angle component = qbA sin (a + <p') 

Q = q cos (a + <p') 25A 
Q = qA cos (a + <p'), 



(91) 



in which A represents the total surface of the thread in actual 
contact. 

The torsional moment of the component at right angles to the 
axis, about the axis, is 



8T 



qd8A 



sin (a + tp') 



Summing up for the entire surface in contact, 
qAd 



T = a^= sin (a + <p') (92) 

Since q and A are generally unknown quantities, it is desirable 



106 EFFICIENCY OF SQUARE THREADS [Chap. IV 

to derive an expression for T in terms of the load Q. This may 
be done by combining (91) and (92), whence 

r = ^tan(a + ^) (93) 

In order to obtain an expression for the efficiency of the square- 
threaded screw, determine the torsional moment To required 
when friction is not considered, and divide this moment by T. 
Without friction <p f = 0; hence, from (93), it follows that 

To = ^ tan a (94) 

Hence, the efficiency is 

T tan a . . 

71 = T = tan (« + *') (95) 

The expression given by (95) could have been obtained directly 
from (79) by making |8 = 90°. 

Very often it is more desirable to have the expressions for T 

and 77 in terms of the coefficient of friction and the dimensions of 

the screw. Letting p represent the pitch of the screw, then 

V 
tan a = — y, also tan <p f = //. Substituting these values in (93) 

and (95), the resulting expression for T is 

r -fE£g] ™ 

and that for the efficiency is 

V ~ rdp - TrVd* (9?) 

The value of the coefficient of friction varies greatly with the 
method of lubrication and the quality of the lubricant. Very 
little experimental information on threads is available; probably 
the most reliable being the results obtained by Prof. A. Kingsbury 
from an extended series of tests on square-threaded screws made 
of various materials, such as mild steel, case-hardened mild 
steel, wrought iron, cast iron, and cast bronze. The results of 
this investigation, some of which are given in Table 24, were 
presented in a paper before the American Society of Mechanical 
Engineers by Prof. Kingsbury, and form a part of volume 17 of 
the Transactions of that society. In the second last column of 
Table 24 are given the mean values of the coefficient of friction 



Art. 88] ' EFFICIENCY OF SQUARE THREADS 107 

Table 24. — Coefficients of Friction for Square-threaded Screws 



Material used for 
screw 


Pres. 

per 

sq. in. 


Material used for nut 


Aver, 
coef. 


Lubricant 
used 


Cast 
iron 


Wrought 
iron 


Mild 
steel 


Cast 
brass 




3,000 


0.1400 
0.1500 
0.1320 
0.1675 
0.1300 


0.1570 
0.1600 
0.1560 
0.1775 
0.1300 


0.1500 
0.1500 
0.1470 
0.1550 
0.1270 


0.1200 
0.1170 
0.1270 
0.1325 
0.1400 


0.143 




Wrought iron 




Mild steel case-hard... 
Cast bronze 


Heavy 

machinery 

Oil 


Cast iron 

Wrought iron 


10,000 


0.1190 
0.1380 
0.1360 
0.1300 
0.1720 


0.1390 
0.1400 
0.1600 
0.1430 
0.1350 


0.1250 
0.1390 
0.1410 
0.1330 
0.1240 


0.1710 
0.1470 
0.1360 
0.1930 
0.1320 


Mild steel case-hard... 








Cast iron 

Wrought iron 


10,000 


0.1050 
0.0750 
0.0650 
0.0700 
. 0440 


0.0710 
0.0700 
0.0675 
0.0550 
0.0450 


0.1075 
0.0890 
0.1110 
0.1275 
0.0710 


0.0590 
0.0550 
0.0400 
0.0350 
0.0360 


0.07 


Heavy 

machinery 

oil 


Mild steel case-hard... 


and 






Cast iron 

Wrought iron 


10,000 


0.0950 
0.1000 
0.1000 
0.1050 
0.1100 


0.1000 
0.1075 
0.1050 
0.0975 
0.1000 


0.1000 
0.1125 
0.1200 
0.1175 
0.1150 


0.1100 
0.1200 
0.1100 
0.1375 
0.1325 


0.11 




Mild steel case-hard... 











for the various lubricants as determined by Prof. Kingsbury, 
and these values are applicable to square-threaded screws run- 
ning at very slow speeds and upon which the bearing pressure 
does not exceed 14,000 pounds per square inch, provided the 
screw is lubricated freely before the pressure is applied. 

(6) Reverse motion. — For the reverse motion of the screw, the 
line of action of the pressure q8A is inclined to the axis of the 
screw at the angle (a — <p') ; hence the moment required to turn 
the screw is 

(T) = ^ tan («-*') 

= Qdr p - vu'd ri 
2Lird + n'pl 



(98) 



For the common screw jack and screws for elevating the cross 
rail on planers, boring mills, and large milling machines, the angle 
of friction <p' exceeds the angle a, thus making (T) negative; that 
is, the lowering of the load Q requires an effort or in other words, 
the screw is said to be self-locking. If, however, <p' < a, the 
moment is positive; that is, an effort must be applied to resist the 
tendency of the load to descend. 



108 



STRESSES IN POWER SCREWS 



[Chap. IV 



From the above discussion, it is evident that in a self-locking 
screw, the limiting value of a is <p f . Substituting a = <p f in (95), 
the maximum efficiency of a self-locking screw is 

_ 1-tonV (99) 



that is to say t] in this case can never exceed 50 per cent. 

89. Stresses in Power Screws. — Screws used for the trans- 
mission of power are subjected to the following stresses: bearing, 
tensile or compressive, and shearing. 

(a) Bearing stresses. — In order that the thread of a screw may 
be capable of transmitting the required power without an undue 
amount of wear, the unit pressure upon the surfaces in contact 
must be kept low, especially if the rubbing speeds are high. In- 
stead of giving this permissible pressure in terms of the normal 
pressure per square inch of actual contact, it is generally quoted 
as so many pounds per square inch of projected area. To deter- 
mine an expression for this quantity in terms of the load on the 
screw, proceed as follows: Using the notation of Art. 88, the 
projected area of the total thread surface in actual contact be- 



tween the nut and its screw is -j- (d\ 



d\), hence 



Q = ™{d\-d\)S, 



(100) 



in which n and S b represent the number of threads in contact and 
the permissible pressure per square inch of projected area, 
respectively. 

The values of Sb given in Table 25 were determined from 
actual screws in service, and may serve as a guide in future 
calculations. 

Table 25. — Bearing Pressures on Power Screws 





Material 


Bearing pressures 


Remarks 




Screw 


Nut 


Min. 


Max. 


Mean 


Jack screw 

Hoisting screw . 
Hoisting screw. 


Steel 
Steel 
Steel 


Cast Iron 

Cast Iron 

Brass 


1,800 
500 
800 


2,600 
1,000 
1,400 


2,200 

750 

" 1,100 


Slow speed 
Medium speed 
Medium speed 



(6) Tensile or compressive stresses. — The method of mounting 
the screw, and the manner of transmitting the desired power, 



Art. 89] STRESSES- IN POWER SCREWS 109 

determine the kind of stress induced in the screw by the action of 
the direct load. The magnitude of this stress is equivalent to 
the load divided by the area at the root of the thread, provided 
the length of the screw if subjected to compression does not ex- 
ceed six or eight times the root diameter. If a screw subjected 
to a compression has a length exceeding the limits just given, 
it must be treated as a column, and the stresses determined ac- 
cording to the formulas given in Art. 15. It is good practice to 
neglect any stiffening effect that the threads may have. 

(c) Shearing stresses. — A torsional or shearing stress is induced 
in the screw by the external turning moment applied, though a 
part of the latter may also be used in overcoming the friction of 
bearings, depending upon the arrangement of the screw and nut. 
In general, the magnitude of the moment causing the shearing is 
never less than that given by (93) or (96), and hence the shearing 
stress induced in this case is 

s. - ^ (101) 

(d) Combined stresses. — -Having determined the magnitude of 
the separate stresses induced in the screw, their combined effect 
must be determined by the principles explained in Art. 17. 



CHAPTER V 

FASTENINGS 

KEYS, COTTERS, AND PINS 

KEYS 

The principal function of keys and pins is to prevent relative 
rotary motion between two parts of a machine, as of a pulley 
about a shaft on which it fits. In general, keys are made either 
straight or slightly tapering. The straight keys are to be pre- 
ferred since they will not disturb the alignment of the parts 
to be keyed, but have the disadvantage that they require accu- 
rate fitting between the hub and shaft. The taper keys by taking 
up the slight play between the hub and shaft are likely to throw 





Fig. 38. 

the wheels or gears out of alignment, but they have the advan- 
tage that any axial motion between the parts is prevented due to 
the wedging action. Keys may be divided into three classes as 
follows : (a) sunk keys ; (6) keys on flats ; (c) friction keys. 

90. Sunk Keys. — The types of sunk keys used most in machine 
construction are those having a rectangular cross-section, though 
occasionally round or pin keys are used. 

(a) Square key. — The so-called square key is only approxi- 
mately square in cross-section and has its opposite sides parallel. 
As shown in Fig. 38(a), this type of key bears only on the sides 
of the key seats, and, being provided with a slight clearance at 
the top and bottom, the key has no tendency to exert a bursting 

110 



Art. 90] 



SUNK KEYS 



111 



pressure upon the hub. To prevent axial movement of the hub, 
set screws bearing upon the key, or other means must be pro- 
vided. The square key is used where accurate concentricity of 
the keyed parts is required, also when the parts must be dis- 
connected frequently, as in machine tools. It is suitable for 
heavy loads, provided set screws are used to prevent tipping of 
the key in its seat. For a list of commercial sizes of square keys 
see Table 29 and Fig. 45(a), to which the dimensions in the 
table refer. 

(6) Flat key. — The flat key has parallel sides, but its top and 
bottom taper. As shown in Fig. 38(6), its thickness t is consider- 
ably less than its width b; furthermore, it fits on all sides, thus 
tending to spring the connected parts and at the same time 
introducing a bursting pressure upon the hub. The fiat key is 
used for either heavy or light service in which the objections just 
mentioned are not serious. 




(a) 



(b) 



Fig. 



(c) Feather key. — The feather key, sometimes called spline, is 
a key fitted only on the sides, thus permitting free axial move- 
ment of the hub along the shaft. Its thickness is usually greater 
than its width, thereby increasing the contact surface' and at the 
same time decreasing the wear. The feather key is fastened to 
either the hub or the shaft, while the key-way in the other part 
is made a nice sliding fit. The key may be secured to the shaft 
by countersunk machine screws or by pins riveted over; or 
when it is desired to fasten the key to the sliding hub, dovetailing 
or riveting may be resorted to. Quite frequently two feather 
keys set 180 degrees apart are used. The stresses are thereby 
equalized, and at the same time it is easier to slide the hub along 
the shaft. 

(d) Woodruff key. — The Woodruff key shown in Fig. 39(a) is 
a modified form of the sunk key. It is patented and is manu- 
factured by the Whitney Mfg. Co. of Hartford, Conn. The key- 



112 



DIMENSIONS OF WOODRUFF KEYS [Chap. V 

Table 26.— Dimensions of Woodhtjff Keys 



No. 


l 


2 


3 


4 


Key 

length 


No. 


l 


2 


3 


4 


Key 

length 


1 

2 


X 


He 

%2 


H* 




V2 


23 
F 


m 


He 

% . 


%2 




IX 


3 




h 








24 




X 

He 








4 




z Ai 








25 


IX 


%4 




IX 


5 


% 


X 

5 A 2 

He 


Vie 




% 


G 




% 








6 
61 


126 
127 
128 


2H 


He 

x 

He 


2 H32 


%2 




7 




X 








1% 


8 


Z A 


%2 

He 
X 


He 




H 


129 




% 








9 
91 


26 

27 
28 


2X 


He 
X 

He 


% 


Xt 




10 




Yz2 








1% 


11 


V* 


He 

7/ 

X 


He 




Vs 


29 




% 








12 
A 


Rx 

Sy 




X 

He 








13 




He 








Tx 


2M 


2 %2 


0.1625 


2 


14 




%2 








Ux 




He 








15 


l 


X He 




l 


Vx 




X 








B 
152 




He 
H 








R 

S 




x 

He 
% 








16 




He 








T 


2M 


% 


X 


2^6 


17 

18 


ix 


%2 

X 


He 4 




IX 


U 
V 




He 
X 








C 




•Tie 








30 




% 

He 








19 




He 








31 










20 




%2 








32 




X 








21 


ix 


x 


%A 




IX 


33 


3M 


He 


l He 


He 


2% 


D 




He 








34 




% 








E 




H 








35 




x He 
S A 








22 


m 


x 


H% 




i% 


36 





seat in the hub is of the usual form, but that in the shaft has a 
circular outline and is considerably deeper than the ordinary 
key-way. ' The extra depth, of course, weakens the shaft, but the 
deep base of the key precludes all possibility of tipping. 

The freedom of the key to adjust itself to the key-seat in the 
hub makes an imperfect fit almost impossible, while with the ordi- 
nary taper key a perfect fit is very difficult to obtain. In secur- 



Art. 90] 



BARTH KEY 



113 



ing long hubs, the depths of the key-way may be diminished by 
using two or more Woodruff keys at intervals in the same key- 
seat. 

In Table 26 are given the stock sizes of Woodruff keys, also 
the various dimensions referred to in Fig. 40. 

To aid the designer in selecting the suitable size of Woodruff 
key for any given diameter of shaft, the information contained 
in Table 27 may be found convenient. 



I, 



i 



-, ? 


[•• ' 


^ 


1 


* \ 


• 

to 1 

» 


t 


r~V^^ 





(a) 



(b) 



Fig. 40. 
Table 27. — Diameters of Shafts and Suitable 'Woodruff Keys 



Shaft 


Key 


Shaft 


Key 


Shaft 


Key 


diam. 


No. 


diam. 


No. 


diam. 


No. 


Ke-% 


1 


%- 15 Ae 


6, 8, 10 


m -iKe 


14, 17, 20 


Ke-K 


2,4 


1 


9, 11, 13 1^-1% 


15, 18, 21, 24 


He-H 


3,5 


nu-iVs 


9, 11, 13, 16 pKe-lM 


18, 21, 24 


%-M 


3,5,7 


lKe 


11, 13, 16 l!% 6 -2 


23, 25 


13 Ae 


6,8 


IK -1%6 


12, 14, 17, 20 2He ~^A 


25 



(e) Lewis hey. — The type of sunk key shown in Fig. 39(6) was 
invented by Mr. Wilfred Lewis. This key is subjected practi- 
cally to a pure compression in the direction of its longest cross- 
sectional dimension, and for that reason the location of this key 
relative to the direction of driving is very important. The Lewis 
key is rather expensive to fit and probably due to that fact is not 
used so extensively, though at the present time one manufacturer 
uses it on large engine shafts. Frequently two such keys are 
used on one hub. 

(/) Barth key. — Some years ago Mr. C. G. Barth invented the 
type of key shown in Fig. 41(a). It consists of an ordinary 
rectangular key with one-half of both sides beveled off at 45 
degrees. With this form of key it is not necessary to make a 
tight fit, since the pressure tends to force the key into its seat. 



114 



KENNEDY KEYS 



[Chap. V 



Furthermore, there is no tendency for the key to turn in its seat, 
since the pressure upon it produces a compression. With re- 
spect to the stresses produced, this key is similar to the Lewis 
key, but has the advantage over the latter that it costs less to 
fit. The Barth key may also be used as a feather key; in many 
cases it has replaced troublesome rectangular feather keys and 
has always given excellent service. 





(b) 



(g) Kennedy keys. — Another system of keying, which has 
given excellent service in heavy rolling-mill work, is shown in Fig. 
41 (6) . This system, known as the Kennedy keys, is similar to that 
in which two Lewis keys are used in one hub. The two keys are 
located in the hub in such a manner that the diagonals pass 
through the center of the shaft as shown in the figure. The 
dimensions of the key at the smaller end are made approximately 
one-fourth of the diameter, of the shaft, and the taper is made 





Fig. 42. 



~% inch per foot. The key should form a driving fit at the top 
and bottom. The following method of fitting a hub with Ken- 
nedy keys represents the practice of a well-known manufacturer, 
and when thus fitted, such keys have always given good results. 
"The hub of the gear after being bored for a press fit with its 
shaft is rebored by offsetting the center approximately %4 inch, 
thus producing the clearance shown in the figure. The keys are 
fitted on the eccentric side of the bore and hence when driven 
home pull the hub into its proper place." The reboring opera- 



Art. 91] 



PIN KEYS 



115 



tion is not essential' to insure good results, but it facilitates 
erection of the parts. 

(h) Round or pin key. — -A round or pin key gives a cheap and 
accurate means of securing a hub to the end of a shaft. This 
form of fastening, shown in Fig. 42(a), was originally intended 
only for light and small work, but if properly designed and con- 
structed will also prove satisfactory for heavy work. The pin, 
either cylindrical or tapering, is fitted halfway into the shaft and 
hub as shown in the figure. For heavy duty, the Nordberg 
Mfg. Co. of Milwaukee uses the proportions given in Table 28, the 
total taper of the reamer being ^6 mcn P er foot. 

For light duty when taper pins are used, it is advisable to 
make use of the so-called "standard taper pins," as they may be 
purchased for less money than it is possible to make them. In 
Table 28 are given the proportions of such pins, also informa- 
tion pertaining to the reamers for these pins. The standard 
taper is % inch per foot. 

Table 28. — Round Keys and Taper Pins 



Nordberg round keys 




Standard taper pins 


ind reamers 






Reamer 


Pins 


Reamer 


Shaft 
diameter 


Small 


Length 


No. 


Large diameter 


Stock 
lengths 


No. 


Small 
diam. 


Length 








of flutes 










Actual 


Approx. 










21^6-3 


?£ 


4K 





0.156 


%2 


%-m 





0.135 


15*6 


3Ke-3H 


Vs 


4K 


1 


0.172 


1 H4 


M-2 


1 


0.146 


lHe 


3%-4 


1 


Ws 


2 


0.193 


He 


s A-m 


2 


0.162 


l^ie 


4^-4^ 


m 


5 


3 


0.219 


J-3 2 


i 


3 


0.183 


2Me 


5 


m 


4% 


4 


0.250 


X 


\ 3 A-3 


4 


0.208 


2% 


5H 


m 


4K 


5 


0.289 


X %4 


J 


5 


0.240 


2V 8 


6 


m 


6H 


6 


0.341 


% 


%-4 


6 


0.279 


3% 


7, 8, 9 


m 


6K,8 


7 


0.409 


l H2 


1 -4 


7 


0.331 


4Kb 


10, 11, 12 


2 


10H 


8 


0.492 


H 


1K-4H 


8 


0.398 


5H 


13, 14, 15 


2% 6 


12 


9 


0.591 


1 %2 


1H-5H 


9 


0.482 


GH 


16, 17, 18 


3H 


13 


10 


0.706 


% 


1H-6 


10 


0.581 


7 


19, 20, 21 


31K6 


















22, 23, 24 


4M 


14K 

















91. Keys on Flats. — A key on the flat of a shaft has parallel 
sides with its top and bottom slightly tapering, and is used for 
transmitting light powers. Fig. 42(6) shows this form of fasten- 
ing. The proportions of keys on flats are about the same as 
those used for the flat key described in Art. 90(6). 



116 



STRENGTH OF KEYS 



[Chap. V 



92. Friction Keys. — The most common form of friction key is 
the saddle key shown in Fig. 43(a), the sides of which are parallel, 
and the top and bottom, slightly tapering. The bottom fits the 
shaft and the holding power of the key is due to friction alone. 
This form of key is intended for very light duty, or in some cases 
for temporary service, as in setting an eccentric. 

93. The Strength of Keys. — Keys are generally proportioned 
by empirical formulas, and in almost all cases such formulas are 
based upon the diameter of the shaft. Neither the twisting 
moment on the shaft nor the length of the key is considered in 
arriving at the cross-section. Since a key is used for torsion 
alone, the twisting moment to be transmitted and not the diam- 
eter of the shaft should fix its dimensions. In the majority 
of cases the shaft must also resist a bending stress in addition 
to the torsional stress, and a larger shaft is required than would 




(a) 




(b) 



be necessary for simple torsion. The empirical formula therefore 
give a larger key than is really needed, thereby increasing the 
cost and at the same time decreasing the effective strength of the 
shaft. The length of the key should be considered in determining 
its crushing and shearing resistance. 

In arriving at the dimensions of the key, the size of the shaft 
should not be disregarded altogether, or the result might be a key 
too small to be fitted properly, or one that is too large. In other 
words, calculate the dimensions of the required key and if neces- 
sary modify these dimensions to suit practical considerations. 
It is generally supposed that keys fail by cross-shearing, but this 
is seldom the case. A large number of failures are due to the 
crushing of the side of the key or key-seat, and for that reason the 
crushing stress should always be investigated. 

(a) Crushing strength. — To determine the crushing stress on 
the side of a key-seat, let T represent the torsional moment 



Art. 94] STRENGTH OF KEYS 117 

transmitted, I the length of the key, and b and t, the dimensions 
indicated in Fig. 43(6). Then the crushing resistance of the 

key is -^r, and its moment about the center of the shaft, whose 

■ Tin 

diameter is d, is approximately . . Equating this moment to 
the torsional moment, and solving for Sb, we have 

Assuming Sb and having given values for T, t and d, (102) may 
be used for calculating the required length of the key. 

Occasionally a key is required to transmit the full power of 
the shaft; hence, making its strength equal to that of the shaft, 
we get 

tldSb = Td s S s 
4 16 ' 

from which 

(b) Shearing strength. — The shearing stress in a key is found 
by equating the torsional moment T to the product of the radius 
of the shaft and the stress over the area exposed to a shear; 
whence 

* " m < 104 > 

Equating the value of T from (102) to that obtained from (104) 
t = 2 6 I s (105) 

Ob 

If Sb = 2 S s , as is generally assumed, (105) calls for a square 
key. To facilitate fitting, the width of the key is frequently 
made greater than its depth, which has the effect of decreasing 
S s relative to Sb. From this it follows that investigations for 
the crushing stress are more essential than those for the shearing 
stress, as in actual practice the latter takes care of itself. 

94. Friction of Feather Keys. — As stated in Art. 90(c), it is 
possible to equalize the pressure coming upon the hub by using 
two feather keys placed 180 degrees apart, thereby reducing 
materially the force required to slide the hub along the shaft. 
The following analysis will serve to show that the statement 
is practically true. 



118 



FRICTION OF FEATHER KEYS 



[Chap. V 



(a) Hub with one feather key. — In Fig. 44(a) is shown a hub 
which is made an easy sliding fit on the shaft and key, the latter 
being fastened securely to the shaft. We shall assume that the 
hub drives the shaft in the direction indicated by the arrow; hence 
the torsional moment T transmitted produces the two forces 
Pi, one of which acts on the key and the other, having the same 
magnitude, causes a pressure on the shaft. These forces being 
parallel form a couple whose moment Pia must equal the tor- 
sional moment T; hence, the magnitude of the force Pi is 



P, = T - 



(106) 



(6) Hub with two feather keys. — In place of a single feather, 
suppose the shaft is equipped with two keys upon which the hub 
slides as shown in Fig. 44(6). Assuming the direction of rotation 





Fig. 44. 

shown in the figure, the forces upon the hub are the two equal 
forces P 2 forming a couple whose moment is 2 P 2 a. Since the 
magnitude of this couple is a measure of the torsional moment T, 
it follows that 

P, = l a (107) 

Comparing (106) and (107), it is quite evident that the force 
producing the frictional resistance in case (6) is only one-half 
as great as that in case (a), assuming the same values of T and 
a, thus showing the advantages gained by the use of two feather 
keys. 

It is important to note that the hub with two feather keys 
requires very accurate fitting in order to produce the action 
assumed in the above analysis. 



Art. 951 



GIB-HEAD KEY 



119 



95. Gib-head Key. — The gib-head or hook-head key is shown 
in Fig. 45, and is nothing more than a flat or square key with the 
head added. This form of key is used in places where it is in- 
convenient or practically impossible to drive out a key from the 
small end. It should be borne in mind, however, that a project- 
ing head is always a source of danger and for that reason 
many engineers condemn its use. In Table 29 are given the 
dimensions of a series of sizes of gib-head keys indicated in Fig. 



1 



f s 


+e 


\( J 


u\« 


;V 


)\ 



(a) 



(b) 



Table 29. — Dimensions of Gib-head Keys 



1 


2 


3 


4 


1 


2 


3 


4 


Vs 


Vs 


Vz2 


K 


Ws 


m 


Ws 


2H 


He 


He 


%2 


He 


iWe 


iWxe 


l 15 Ae 


2% 


H 


y± 


% 


W32 


w 


w 


2 


3 


He 


He 


W S 2 


He 


iWxe 


lWxe 


2Kb 


Ws 


% 


Vs 


15 A 2 


We 


Ws 


Ws 


2\i 


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Ws 


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Vs 


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1H 


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4 


We 


We 


2 %2 


We 


2He 


2He 


2He 


Ws 


% 


y± 


Vs 


w 


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2K 


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4K 


l He 


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We 


We 


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m 


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Ws 


2Vxe 


2Vxe 


2Wxe 


4^8 


1 


1 


Ws 


W 


2y 2 


2V 2 


2% 


4% 


1/16 


We 


We 


lWxe 


2He 


2He 


2Wxe 


m 


Ws 


w 


We 


Ws 


2% 


2% 


3 


5 


We 


We 


W 


lWxe 


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5 


m 


w 


We 


2 


2% 


2M 


Ws 


Ws 


We 


We 


w 


2M 


2^He 


2% 


Wxe 


w% 


w 


w 


We 


2H 


2% 


2% 


Wa 


W± 


We 


We 


W 


2% 


2^16 


2Wxe 


Wxe 


• W 


W 


W 


w 


2\i 


3 


3 


w 2 


Ws 


We 


We 


iWe 


2% 











120 SHAFT SPLINES [Chap. V 

45(a). The keys listed in this table are square in cross-section 
at the head end, and have a taper of % mcn P er f°°t- 

96. Key Dimensions. — In Fig. 45(6) are shown the dimensions 
that will prove most convenient for the shop man in order to 
machine the key-seats in the hub and shaft. The dimension a 
is the one used for arriving at the proper depth of the key-seat 
in the hub. To arrive at the depth of the key-seat in the shaft, 
the majority of the workmen prefer to have given the dimension 
c, as that is by far the most convenient dimension when the key- 
seat is cut on a milling machine. Some mechanics prefer to use 
the dimension e in place of c thus enabling them to use calipers. 

97. Integral Shaft Splines. — With the development of the auto- 
mobile, the defects of the inserted keys in circular shafts became 
apparent, and finally the old key construction was discarded 
almost altogether, in particular in the sliding-gear construction 
and rear-axle transmissions. Due to the weakening of the shaft 
by the inserted key, the square shaft was at first introduced, and 
this met with considerable success. The square shaft, however, 
is considerably heavier than a circular shaft of the same strength, 
so in order to keep the weight down and at the same time provide 
greater key-bearing area, the automobile designer developed what 
is now called the integral spline shaft. Such a shaft is simply 
a round shaft in which the splines are produced by milling out 
the metal between them. 

At first the integral spline shafts were produced on the milling 
machine, but at present they can be produced more cheaply on 
the hobbing machine. The splined holes through the hubs of the 
gears which slide over such shafts are produced very accurately 
and cheaply on a broaching machine. It is claimed by some 
manufacturers that the cost of hobbing a multiple-spline shaft 
and broaching the hub to fit the shaft is considerably less than 
the combined cost of turning the circular shaft, cutting the key- 
way in it, boring the gear to fit the shaft, cutting the key-way in 
the gear, and fitting the key. 

The automobile manufacturer is not the only one that is using 
integral spline shafts; the advantages of such shafts are so appar- 
ent that a considerable number of machine tool builders are now 
using them in connection with their sliding change-gear mechan- 
isms. As now used in the various classes of service, the integral 
spline shafts are constructed with from four to ten splines. In 



Art. 97] 



SHAFT SPLINES 



121 



Fig. 46(a) and (o) are shown the cross-sections of a hub contain- 
ing six and ten splines respectively, the former being used for 
the sliding gears, while the latter is applied to the rear axle. The 
proportions of the two types shown in Fig. 46 have been standard- 
ized by the Society of Automobile Engineers. Each of these types 
is made in three different sizes, A, B and C, and the following 
formulas give the dimensions of the various parts of the bore, 
while the corresponding parts of the shaft are made one-thou- 
sandth of an inch less on the smaller shaft diameters and two 
one-thousandths on the larger sizes. 

For the six-spline type shown in Fig. 46(a), the formula for the 
width b of the spline is the same for all three sizes; the other di- 
mensions, however, vary. 





(108) 



For Q-A, d = 0.90 D 

b = 0.25 D 
t = 0.05 D 
For G-B, d = 0.85 D 

t = 0.075 D 
For G-C, d = 0.80 D 

t = 0.10 D 

As in the ease of the six splines, the width b for the three sizes 
of the ten-spline fitting shown in Fig. 46(o) is kept constant. 
The various proportions are given by the following formulas : 



For 10-A, 



For 10-5, 



For 10-C, 



i = 0.91 D 
b = 0.156 D 
t = 0.045 D 
I = 0.86 D 
t = 0.07 D 
i = 0.81 D 
t = 0.095 D 



(109) 



122 



COTTER JOINTS 



[Chap. V 



COTTER JOINTS 

A cotter is a cross-key used for joining rods and hubs that are 
subjected to a tension or compression in the direction of their 
axis, as in a piston rod and its cross-head ; valve rod and its stem ; 
a strap end and its connecting rod. 

98. Analysis of a Cotter Joint. — In Fig. 47 is shown one method 
of joining two rods through the medium of a cotter, the rod being 
loaded axially as shown. The joint may fail in any one of the ten 
ways discussed below. 

(a) Rods may fail in tension. — The relation between the 
external force P and the internal resistance of the rod is given by 
the following formula: 

p Td*S t 
4 



(HO) 




Fig. 47. 



(b) Failure of the rod across the slot. — Equating the external 
force to the tension in the rod across the slot, we get 



P = 



[¥-«.]*. 



(in) 



(c) Failure of the socket across the slot. — Equating the external 
force to the internal resistance due to the tension in the socket 
across the slot, we find that 



P =[l(D* - dl)-(D - dJ^St 



(112) 



(d) Cotter may shear. — Due to the force P, the cotter may 
fail by double shearing ; hence the relation between the load and 
stress is as follows : 

P = 2btS s (113) 

0) Rod end may shear. — To prevent the rod end from failing 



Art. 



COTTER JOINTS 123 



due to double shearing through the length a, the following ex- 
pression may be used to determine the minimum value of a: 

P = 2ad 1 S s (114) 

(/) Socket end may shear. — The dimension c must be made 
long enough so that the end of the socket will not fail by double 
shearing. Equating the internal resistance to the force P, we 
get 

P = 2c(D - di)S s (115) 

(g) Socket or cotter may crush. — The external force may crush 
either the cotter or the socket along the surfaces AB and CE; 
hence, liberal surfaces must be provided. The following expres- 
sion gives the relation between the load and stresses : 

P = t(D-d i )S b (116) 

(h) Rod or cotter may crush. — To prevent the rod or cotter 
from crushing along the surface FG, the relation expressed by the 
following formula must be fulfilled: 

P = tdtS b (117) 

The cotter joint illustrated by Fig. 47 may also be used for a 
class of service in which the force P may be reversed in direction, 
thus producing a compression in the rod in place of a tension. 
Such a loading will then call for an investigation of the collar. 

(i) Collar may shear off. — Due to the compression in the 
rods, the collar may shear off; whence 

P = Td ie S s (118) 

( j) Collar may crush. — To prevent crushing of the collar, the 
surface in contact must be made large enough so that the follow- 
ing relation between the load and stress is satisfied : 

P = I (dl - dl)S b (119) 

The taper on the cotter should not be made excessive, or 
trouble may be experienced due to the loosening of the cotter 
when the joint is under load. To prevent such loosening, the 
cotter may be provided with a set screw. A practical taper is 
}4, inch per foot, but this may be increased to* 1% inches per 
foot, provided some locking device is applied to the cotter. The 
cotter instead of being made square-ended as shown in Fig. 47, 



124 



TAPER PINS 



[Chap. V 



is more often made with semi-circular edges. This method of 
making the cotter possesses the following advantages: 

1. Sharp corners that are liable to start cracks are avoided. 

2. The shearing area at the sides of the slots is increased con- 
siderably. 

3. The slots with semi-circular ends cost less to make. 

PIN JOINTS 

In Art. 90(h), the use of round and taper pins in the form of 
keys was discussed, and in the following articles additional uses 
of pins will be taken up. These uses are as follows: 

(a) For rigid fastenings in which the pins are so placed, that 
they are either in single or double shear due to the external force. 

(b) For joining two rods which require a certain amount of 
motion at the joint. 

99. Taper Pins. — Taper pins properly fitted form a cheap and 
convenient means of fastening light gears, hand wheels and levers 




(b) 



Fig. 48. 



to shafts that transmit a small amount of power. They may 
also be used for making a connection between two rods, similar 
to the cotter joint described in Art. 98. The common method of 
applying taper pins is illustrated in Fig. 48(a) ; but this method is 
applicable to the transmission of a torque in only one direction. 
If the machine parts are subjected to alternating stresses, as 
would be the case in a coupling between the valve rod and the 
valve stem, the taper pins should be given a slight clearance 
similar to that provided for the cotter in Fig. 47. 

Another very important application of taper pins is their use 
as dowel pins on bearing flanges, and all forms of brackets and 
attachments on machine frames. The main function of dowel 



Art. 100] TAPER PINS 125 

pins is to form a convenient means of locating accurately a 
bearing or bracket, since cap screws and studs cannot be relied on 
for that purpose. If the taper pins are fitted correctly and 
located properly, no trouble is experienced in reassembling the 
machine parts after being dismantled for repair or other purposes. 
In Fig. 36 is shown the application of two taper dowel pins c on 
the flange of a solid bearing. It should be observed that these 
pins are not diametrically opposite, though in this case they 
could have been located symmetrically, since the location of the 
oil hole in the bearing would insure the correct assembling. 
However, many symmetrical castings or brackets are used, and 
the location of the dowel pins as illustrated in Fig. 36 may obviate 
a lot of unnecessary work. Another function of dowel pins, 
which in many cases is of great importance, is to make these pins 
take the shearing action due to the external load, thus relieving 
the cap screws or studs from such action. 

As mentioned in Art. 90 (h), standard taper pins cost but 
little, and the various sizes and lengths listed in Table 28 are 
carried regularly in stock by the various manufacturers. The 
taper adopted by the manufacturers is one-fourth inch per foot. 
These standard taper pins have no provision on the head or point 
that will allow for upsetting the ends, if desired. Provisions 
for upsetting can be made by having the heads and points tapered, 
which would also facilitate the driving of the pin into the machine 
part as well as its removal. For removing large dowel pins such 
as are used in locating the housings on planers and heavy milling 
machines, the taper pin is provided at the large end with a 
threaded shank which is fitted with a nut; hence to remove the 
dowel pin merely back out the pin by screwing up the nut. 

Occasionally a threaded shank is provided at the small end of 
the pin, which if fitted with a nut forms an effective means of 
retaining a pin having a steep taper. When the taper pin is 
used as a fastening similar to that shown in Fig. 48(a), the large 
diameter D of the pin is made from one-fourth to one-third of the 
diameter of the rod or shaft through which the pin passes. The 
length L Fig. 48(6) is chosen so that the pin projects a small 
amount on each side of the hub, though not enough to make it 
dangerous. Table 28 also contains information pertaining to the 
standard reamers that are used with the standard taper pins. 

100. Rod and Yoke Ends. — Various forms of pin joints are 
used for connecting together two or more rods and at the same 



126 



ROD AND YOKE ENDS 



[Chap. V 




r 




CO 




H^ 



an 



— 


-3 — 

'1 


"" 1 

Ok 


— 


— 




d—- 


/f" 






Fig. 50. 



Art. 100] 



ROD AND YOKE ENDS 



127 



time permitting a certain amount 
of motion at the joint. Such joints 
are called rod and yoke ends or 
knuckle joints and are used in prac- 
tically all classes of machinery. In 
Fig. 49 are shown the standard 
drop-forged rod and yoke ends 
adopted by the Society of Auto- 
mobile Engineers, and the propor- 
tions thereof are included in Table 

30. It should be noticed that the 
yoke ends are made in two types, 
namely, the adjustable and the 
plain, illustrated by Fig. 49(a) and 
(6), respectively. 

The sizes of yoke and rod ends 
used in the automobile industry do 
not cover a wide range, and in 
order to meet the demand for yoke 
and rod ends adapted to general 
use, several manufacturers of drop 
forgings carry such parts in stock. 
In Fig. 50 are shown finished plain 
rod and yoke ends that are a 
standard product of The Billings 
and Spencer Co. of Hartford, 
Conn. The dimensions indicated 
in Fig. 50 are included in Table 

31. The plain shanks of these 
forgings are made of sufficient 
length to permit welding them on 
to rods of any desired length. 

The type of rod end just dis- 
cussed has no provision whatever 
for taking up wear at the joint, and 
in the class of service for which 
they are intended, it is not custo- 
mary to make such provision. 
There are, however, many places 
where the wear on the pin or its 
bearing must be taken up and 



c 

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>> 

a 
1 
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CO CN CN CM CM CM 


t> 


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128 



ROD AND YOKE ENDS 



[Chap. V 



hence the design of such rod ends requires some knowledge of 
bearing and journal design. Such machine parts are discussed 
in detail in Chapter XIX. 





Table 31- 


-B. & S. Drop-forged Rod and Yoke Ends 








Rod end 






Yoke end 






No. 


d 






1 












1 


2 


3 


4 


i 


2 


3 


4 


5 





k 


3% 


% 


Ke 


He 


4K 


2 %2 


K 


Ke 


Ke 


1 


Ke 


4^6 


ls Ae 


% 


Ke 


4K 


% 


K 


K 


Ke 


2 


% 


4M 


K 


Ke 


% 


5 


1Kb 


l 


Ke 


K 


3 


Ke 


4Kb 


1 


y 2 


Ke 


5K 


1^2 


IK 


K 


Ke 


4 


k 


4% 


1H 


Vie 


K 


5K 


IKe 


IK 


Ke 


K 


5 


He 


4^6 


IK 


K 


Me 


5K 


IKe 


IK 


K 


Ke 


6 


% 


5^6 


m 


k 


% 


6KJ 


1% 


IK 


K 


K 


7 


k 


5%6 


m 


% 


K 


7 


1^6 


IK 


K 


K 


8 


XL. 


'6%2 


2 


i 


K- 


7K 


2K4 


2Ke 


1 


K 


9 


1 


-6K 


2K 


1% 


l - 


8V 2 


2% 


2K 


IK 


1 


10 


IK 


7^2 


2% 6 


IK 


IK' 


9K 


2 2 %2 


2% 


IK 


IK 


11 


IK 


7%6 


2% 


1M 


IK . 


10 


3%2 


3K 


IK 


IK 



References 



Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 
Mechanical Engineers' Handbook, by L. S. Marks, Editor in Chief. 
Kent's Mechanical Engineers' Pocket Book. 



CHAPTER VI 

CYLINDERS, PLATES AND SPRINGS 
CYLINDERS 

In the following discussion, cylinders will be divided into two 
general classes, as follows: 

(a) Those having thin walls, as for example, steam and water 
pipes, boiler shells and drums. 

(6) Those having relatively thick walls. 

101. Thin Cylinders. — In analyzing the stresses induced in the 
walls of thin cylinders by an internal pressure, we shall assume, 
first, that the stresses are distributed uniformly over the cross- 
section of the cylinder; and second, that the restraining action of 
the heads at the ends of the cylinder is zero. Considering a 
cylinder having its ends closed by heads, the internal pressure 
against these heads produces a longitudinal stress in the walls; 
the magnitude of which is 

S t = f> (120) 

in which d represents the inner diameter of the cylinder, p the 
unit internal pressure and t the thickness of the cylinder walls. 

Assuming that the above cylinder is cut by a plane through its 
axis, the resultant internal pressure on a section of either half 
cylinder having a length L as pdL; hence, the magnitude of the 
tangential or hoop stress is 

s: - f t (12D 

Comparing (120) and (121), it is apparent that the longitudinal 
stress S t is one-half of the tangential stress; however, the true 
tangential stress is even less than that given by (121). Assum- 
ing that Poisson's ratio has a value of 0.3, the effective tangential 
stress is 

„ pd 0.3 pd 0.462 pd 

St= 2t~ ~T7~ - t (122) 

Designers never use formula (122); they prefer (121) since the 
129 



130 



THICK CYLINDERS 



[Chap. VI 



thickness of the walls obtained by the latter, for any assumed set 
of conditions, is always greater. 

102. Thick Cylinders. — In a cylinder having walls that are 
thick when compared to the internal diameter, the stresses in- 
duced by an internal pressure p cannot be considered uniformly 
distributed as in the preceding case. The tangential stress, or 
hoop tension as it is frequently called, varies along the wall 
thickness, having its greatest magnitude at the interior of the 
cylinder and its minimum at the exterior surface. Several 
investigators have proposed formulas that are applicable to the 

design of thick cylinders, among 
the most prominent of these be- 
ing Lame, Clavarino, and Birnie. 
(a) Lame's formula. — In the 
case of a cylinder subjected to 
both internal and external pres- 
sure, as shown in Fig. 51, the 
tangential and radial stresses at 
the variable radius r are given, 
according to Lame, by the follow- 
ing expressions: 

Si = M + - 2 (123) 




in which 



and 



M 



N = 



N 

S r = M - ~ 2 > 



>d 2 - p D°< 
D 2 - d 2 



d 2 D' 



X V ~ 7>o "| 
ID 2 - d 2 \ 



(124) 
(125) 
(126) 



In order to derive a formula that is applicable to thick cylinders 
subjected only to internal pressure, we make po = in (125) and 
(126); then the maximum tangential stress occurs on the inner 
surface of the cylinder, and its magnitude is 



[D 2 + d 2 l 



(127) 



This is one of the forms in which the Lame formula may be 
used, but very often it is found that another form is more con- 



Art. 102] BIRN1KS FORMULA 131 

venient. This may be derived by clearing (127) of fractions and 
substituting (2 t + d) for D, whence 



=ium-i] 



'-ibim- 1 ] (128) 

(b) Clavarino's and Birnie's formulas. — In the preceding dis- 
cussion, Poisson's ratio of lateral contraction was not intro- 
duced, and for that reason (127) and (128) are only approximate. 

According to the maximum-strain theory proposed by Saint 
Venant, the effective tangential and radial stresses are as follows: 

IZeI 1 ^ 

in which 8 t and 8 r represent the unit tangential and radial strains. 
It is evident that these strains or deformations depend on the 
longitudinal stress in the walls of the cylinder. Two cases may 
occur, namely, a cylinder may have its ends open or the ends may 
be closed. 

1. Cylinder with open ends. — In a cylinder having open ends, 
the longitudinal stress is zero; and assuming the cylinder to be 
under an internal pressure, the maximum tangential stress is 

S t = (1 - m) M + (1 ■+' hi) ^ (130) 

in which m represents Poisson's ratio. 

Substituting the values of M and N from (125) and (126) in 
(130), we get finally 

S t = D2 P _ d2 [(1 - ni)d> + (1 + m)D*] (131) 

Substituting in (131), the value of D in terms of d and t, we 
have 

t=-\JV^\ 1 ~ m l P -l\ (132) 



= d [ j S t + (1 - m) y _ A 
2 [\IS t - (1 + m) p J 



This formula is that due to Birnie and applies only to cylinders 
having open ends. 

2. Cylinder with closed ends. — The second case mentioned 
above is the one of most frequent occurrence, namely, that in 
which the ends of the cylinder under internal pressure are closed. 
For this condition, the magnitude of the maximum effective 



132 CLAVARINO'S FORMULA [Chap. VI 

tangential stress is given by the expression 

S t = (1 - 2m) M + (1 + m) ^ (133) 

from whkh 

8 *' = D^-SP- [(1 ~ 2 W) d2 + (1 + m) D2] (134) 

If an expression for t is desired, it may be obtained from (134) 
by substituting for D its value in terms of t and d; whence 



,d\ I S t +{l-2m)p 1 ( . 

1 ~ 2 [\ S - (1 + m) V ~ J (135) 

This expression is known as Clavarino's formula and applies 
to all cylinders, under internal pressure, having closed ends. 

For values of m to be used in the above formulas, refer to 
Table 1. 

PLATE 

The various formulas in common use for determining the 
strength of flat plates subjected to various methods of loading 
are generally based upon some arbitrary assumption regarding 
the critical section or the reactions of the supports. Grashof, 
Bach, Merriman, and others have treated this subject from a 
mathematical standpoint, and the various formulas proposed by 
these investigators give results that agree fairly well with the 
experimental results obtained by Bach, Benjamin, Bryson, and 
others. Flat plates subjected to various methods of loading are 
of frequent occurrence in machines, and the formulas in the 
following articles are those proposed by Prof. Bach. They 
are reliable and are comparatively easy to apply to any given 
set of conditions. It should be understood, however, that these 
formulas apply only to the plain flat plate and not to plates hav- 
ing a series of reenforcing ribs that are commonly used when the 
plates are cast. 

103. Rectangular Plates. — In arriving at formulas for the 
strength of rectangular plates, the critical section is taken as pass- 
ing through the center of the plate, and the part to one side of 
this section is treated as a cantilever beam. The location of the 
critical section is determined by experiments, and for rectangular 
plates made of homogeneous material, Bach found that failure 
does not always occur along a diagonal as in the case with square 



Art. 104] 



RECTANGULAR PLATES 



133 



plates. However, in establishing a general formula, it is usually 
assumed that the line of maximum stress lies along the diagonal. 
(a) Uniformly loaded. — Consider a rectangular plate of 
thickness t, of length a and of breadth b, as supported at the 
periphery and subjected to a pressure p that is uniformly dis- 
tributed; then, according to Bach, 



pK 



°["-m 



(136) 



in which K is a coefficient depending upon the method of sup- 
porting the periphery of the plate, the condition of the surface of 
the plate, the initial force required to make a tight joint, and the 
material used for making the tight joint. The values of K for 
cast iron and mild steel for various conditions of supporting the 
loaded plate are given in Table 32. 



Table 32. — Values op Cos 


PFICIENl 


s K 


, Ki, K 2 AND K 3 




Material 


Condition of 

support 


K 


Ki 


K 2 


K 3 


Cast iron i 


Free 

Fixed 

Free 

Fixed 


0.565 
0.375 
0.360 
0.240 


2.6-3 





0.282 
0.187 
0.180 
0.120 


1.3-1 


5 




Mild steel 







(b) Central loading. — Suppose now that a rectangular plate 
having the same dimensions as the one discussed previously be 
supported at the periphery and loaded centrally by a load Q; then 
the thickness may be determined by the following expression: 



-4 



abQKi 



S (a 2 + 6 2 ) 



(137) 



For a cast-iron plate supported freely, the value of K x as deter- 
mined experimentally by Bach varies from 2.6 to 3.0. 

104. Square Plates. — For similar conditions of loading and 
supporting the plate, the formulas for the thickness of square 
plates may be derived directly from the corresponding formula 
pertaining to rectangular plates. Therefore, for uniformly dis- 
tributed pressure, the thickness is 



-«>/*•! 



(138) 



134 



CIRCULAR PLATES 



[Chap. VI 



For a square plate centrally loaded, it is 



(139) 



For values of K 2 and K 3 , consult Table 32. 

105. Circular Plates. — (a) Pressure uniformly distributed. — The 
thickness of a circular plate having a diameter a, and which is 
supported around its circumference and subjected to a uniformly 
distributed pressure, is determined by the following formula: 



a-JKi 



and its deflection is given by the expression 

a'pK, 



A = 



Et 3 



(140) 



(141) 



In (140) and (141), i? 4 and K 5 represent coefficients which 
depend upon the method of support as well as the method and 
materials used in making the joint tight. Values of these 
coefficients are given in Table 33. 



Table 33. — Values of Coefficients K 


4, K 5 , K 6 AND K 7 


Material 


Condition of 
support 


Ki 


Kt 


Ke 


A> 


Cast iron-! 
Mild steel I 


Free 

Fixed 

Free 

Fixed 


0.30 
0.20 
0.19 
0.13 


0.038 
0.010 


1.43 


0.1-0.125 



(b) Central loading. — For a flat circular plate supported freely 

around the circumference and subjected to a load Q at the center 

ird 2 
which is considered as distributed uniformly over the area -r^> 

the thickness is given by the following formula : 



VM 1 -IS 



2d-] Q 

S 



(142) 



The deflection caused by the load Q may be determined by the 
relation 

A a 2 QK 7 



Et 3 



(143) 



For values of the coefficients K 6 and K 7 , consult Table 33. 



Art. 106] 



CYLINDER HEADS 



135 



According to Grashof, the thickness of a circular plate fixed 
rigidly around the circumference and loaded centrally by the 
load Q may be calculated by the relation 



/0.435 Q . a 

S—s~ loge d 



(144) 



If the deflection is desired, the following expression may be used: 
0.055 Pa 2 



Et s 



(145) 



106. Flat Heads of Cylinders. — (a) Cast heads. — In the case 
of a cast-iron cylinder having the flat head cast integral with the 
sides, as shown in Fig. 52, the allowable stress in the head, 
according to Bach, is given by the relation 



S = 0.8 p 



(i+V) 



2t 



(146) 



I 



^sW^yvsN^ gpj 



(a) 



[b) 



Fig. 52. 



i 



(6) Riveted heads. — The stress in the flat head riveted to a 
cylindrical shell, according to Bach, is 



S = p 



2, + 0.38 



-'(* + t) 



2t 



(147) 



in which the various symbols have the same meaning as above. 

107. Elliptical Plates. — (a) Pressure uniformly distributed. — 
Plates having an elliptical form are frequently met with in en- 
gineering designs; for example, handhole plates and covers for 
manholes in pressure vessels. The following formula, due to 
Bach, gives the thickness of an elliptical plate subjected to a 



136 



HELICAL SPRINGS 



[Chap. VI 



uniformly distributed pressure, and whose major and minor 
axes are a and b, respectively: 



K*b 



t'+m 



(148) 



Table 34. 



The values of K s for cast iron and mild steel, and for two condi- 
tions of supporting 
-Values of Coefficients K 8 and K 9 the pkt6j are giyen 

in Table 34. 

(b) Central load- 
ing. — The thickness 
of an elliptical plate 
supported around 
the periphery and 
subjected to a load 



Material 


Condition of 
support 


K s 


K s 


Cast iron. j 
Mild steel I 


Free 

Fixed 

Free... 

Fixed 


0.82 
0.58 
0.60 
0.46 


0.85 
0.77 



Q at the center is given by the following expression: 



\ A9 L3 + 2c 2 + 3c 4 J 



S 



(149) 



2 c 2 + 3 . 

in which c represents the ratio of the minor axis b to the major 
axis a. For values of Kg, for various conditions of loading, con- 
sult Table 34. 

SPRINGS 

Springs are made in a variety of forms, depending upon the 
class of service for which they are intended. Among the com- 
mon forms used to a considerable extent in connection with 
machinery, are the following: (a) Helical springs; (6) spiral 
springs; (c) conical springs; (d) leaf springs. 

108. Helical Springs. — Helical springs are used chiefly to resist 
any force or action that tends to lengthen, shorten, or twist them. 
The wire or bar used to make this type of spring may have a cir- 
cular, square, or rectangular cross-section. The stresses induced 
in the material of a helical spring subjected to an extension or a 
compression consist of a tension combined with secondary stresses, 
such as tensile and compressive due to a bending action. The 
latter stresses are generally not considered in the development of 
suitable formulas for the permissible load and the deflection. 

(a) Circular wire. — The method of procedure in arriving at 



Art. 108] HELICAL SPRINGS 137 

the relations existing between the axial deflection and the axial 
load for a helical spring made of round wire is as follows : 

Let D = mean diameter of the coils. 
E s = torsional modulus of elasticity. 
Q = axial load on the spring. 
d = diameter of the wire. 
n = number of coils. 
p = pitch of coils. 
A = total axial deflection. 

The stresses at any section of the bar at right angles to the axis 

of the spring are those due to the torsional moment —~- and to the 

bending action, the effect of the latter being disregarded. Apply- 
ing the formula for torsional stress from Art. 10, we have 

S. = ^ (150) 

7TGT 

In determining the safe stress for any given case by means of 
(150), the magnitude of Q must be taken as the greatest load that 
will ever come upon the spring. Frequently (150) is used for 
calculating the safe load that a spring will carry, or it may be 
used for arriving at the size of the wire required for a given 
load, safe working stress, and diameter of coil. 

The total length of the bar required to make the spring is 
nVVD 2 + p 2 or approximately irnD, and according to Art. 10, 
the angular deflection of a bar having the length just given, is 

„ 360 nDSs 



dE s 



(151) 



The axial deflection of the spring is evidently given by the 
following formula: 

irnD 2 S s /., e \ 

A - --mr im 

Substituting the value of S s from (150) in (152) 

Having given the load Q and the corresponding deflection 
A, (153) will be found useful for determining the required number 
of coils n, by assuming values for the size of wire and the diameter 



138 



HELICAL SPRINGS 



[Chap. VI 



of the coils. The formula for the deflection as given by (152) 
may be used for calculating the safe deflection. 

In designing helical springs, the following method of procedure 
is suggested : 

1. By means of (150), determine the diameter of the coil re- 
quired for the given load and assumed values of the fiber stress 
and size of wire. The results obtained may have to be rounded 
out so as not to get an odd size of arbor upon which the spring 
is made. 

2. Having arrived at a proper dimension for the diameter of 
the coil, the deflection may be determined by means of (153), 
provided we know the number of coils required, or if the deflec- 
tion is fixed by the surrounding conditions, the number of coils 
required may be calculated by means of (153). 

(b) Bar having rectangular cross-section. — For helical springs 
made of a wire or bar having a rectangular cross-section b X h, 
as shown in Fig. 53, the relation between the fiber stress S s and 




Fig. 53. 



the external load Q is obtained by equating the external moment 
to the moment of resistance; whence 



S s = 



9QD 

4 6 2 A 



(154) 



This formula is used to establish the size of the wire for any 
given load and safe stress, or it may be used to check the stress 
having given the load and size of wire. 

According to the Mechanical Engineers' Handbook, the axial 
deflection of the spring may be calculated by the following 
formula : 

_ 2.83nQ£>3 (62 + /t2) . 

A ~ VVE S (155} 

If an expression for the axial deflection is desired in terms of the 



Art. 109] HELICAL SPRINGS 139 

safe stress S„, the value of Q obtained from (154) is substituted 
in (155); whence 

1.26 D*S S (6 2 + h>) 
A = bh*E s (156) 

The method of procedure to be used in the design of a helical 
spring constructed of a rectangular bar, as shown in Fig. 53, 
is the same as that suggested in (a) above. 

(c) Bar having square cross-section. — In many installations 
requiring helical springs, square wire is preferred to the rectangu- 
lar. By making b = h in (154), (155) and (156), we obtain the 
desired equations necessary for designing springs constructed of 
square wires. 

For a given load and assumed fiber stress, the size of the wire 
or bar may be calculated by means of the following formula: 

Q 9QD Mm 

S s = -j^- (157) 

The axial deflection may be determined from ^ 

A _ 5^ (158) 

or from 

A - ^|p (159) 

For the method of procedure, the suggestions given in (a) 
above may be followed. 

109. Concentric Helical Springs. — The springs used in many 
automobile clutches, as well as those used on railway trucks, 
consist of two concentric helical coils, both of which are neces- 
sarily deflected equal amounts, since their free and solid lengths 
are made equal. The springs used on railway trucks are gener- 
ally made of round bars, while those used for automobile clutches 
are made of round, rectangular and square stock. In actual 
construction, the adjacent coils of concentric springs are wound 
right and left hand so as to prevent any tendency to bind. In 
the design of concentric springs in which the same grade of 
material is employed, an attempt should be made to get ap- 
proximately the same stresses in the various coils. With the 
use of round wire, the latter condition is met by making the ratio 

-j the same for all coils, as the following analysis shows : 



140 HELICAL SPRINGS [Chap. VI 

Using the same notation as before and representing the solid 
length of the spring by H, but adopting the subscripts 2 and 1 
to the various dimensions of the inner and outer coils respectively, 
it follows from (152) that the stress in the material of the inner 
coils of a double helical spring is 

and that in the outer coils 

Now, assuming that the deflections and the solid heights are to 
be the same for the two coils, it is evident that for equal stresses 

L\ = L\ 

di d,2 

Since the ratio -3 is the same for both coils, it follows that the 
a 

lengths of the bars from which the separate coils are made will 

be the same. 

110. Helical Springs for Torsion. — Helical springs are also 
used to resist a torsional moment T by having one end held 
rigidly while the other is relatively free. Such springs are in- 
variably made from bars having a rectangular or square cross- 
section. The material of the spring is subjected to a bending 
stress having a magnitude as follows: 

s = m w 

in which h is the width and b the radial thickness of the spring 
stock. 

The linear deflection according to the Mechanical Engineers' 
Handbook is 

. TLD LSD 

A =2M = ~ltt' (163) 

in which the total length L of the bar may be assumed equal to 
irnD, as in Art. 108(a). 

For springs made of square wire, the formulas for stress and 
deflection may be derived from (162) and (163) by making h = b. 

111. Spiral Springs. — The spiral spring is used but little in 
machine construction, and then only for light loads. It consti- 



Art. 112] 



CONICAL SPRINGS 



141 



tutes what is commonly called a torsional spring and the material 
used in its construction is subjected to a bending stress. Letting 
h represent the width and b the radial thickness of the spring 
material, the moment of the external force Q must equal the 
internal resistance; hence 

« 3 QD 



(164) 



The following expression for the linear deflection A of a spiral 
spring is that given in the Mechanical Engineer's Handbook. 

QLD 2 = LSD 
4: EI bE' 



A = 



(165) 



in which L represents the length of the straightened spring and 

the other symbols are as in the preceding 

articles. 

112. Conical Springs. — Conical springs 
are generally used to resist a compression 
and are made of round or rectangular stock. 
They are applicable where the space is 
limited, and where there is no necessity for 
great deflections. The following formulas 
derived from the Mechanical Engineers' 
Handbook may serve for determining the 
proportions of such springs: 

For a conical spring made of round stock 
and loaded as shown in Fig. 54, the shear- 
ing-stress in the material is as follows: 




8, 



8QD, 



(166) 



The axial deflection for n turns or coils is given by the following 
expression : 

. _ 2wQ /rk 3 , ^2^ , r,2^ i r^3> 



d*E t 



(Dt + DWj + DlD t + Dt) 



(167) 



If the expression for the deflection is desired in terms of the 
safe stress, we have 



■nS s 



4 dD 2 E s 



(Dl + D\D X + DlD 2 + D{) 



(168) 



A conical spring made of rectangular stock is shown in Fig. 55. 



142 



LEAF SPRINGS 



[Chap. VI 



The torsional stress in the material of such a spring may be 
calculated by the formula 



9QD 2 

4b 2 h 



(169) 




The axial deflection in terms of the load Q is 

0.71 nQ (V + h>) (Dl + D\ D X + D*D\ + D\) 
b 3 h*E s 
In terms of the safe stress, the axial deflection is 

0.315 n (6 2 + A 2 ) (Dl + D\D X + D 2 D\ + Z)J) 



A = 



bhW 2 E s 



(170) 



(171) 




Fig. 56. 



113. Leaf Springs. — Leaf springs are made in various forms 
some of which are shown in Fig. 56. The first form shown is 
called the full elliptic, the second semi-elliptic and the ordinary 



Art. 114] SEMI-ELLIPTIC SPRINGS 143 

flat leaf spring is represented by Fig. 56(c). In all of the forms 
shown, the various leaves are banded tightly together, and, as 
usually constructed, each type has one or more full-length leaves, 
sometimes called master leaves, while the remaining leaves are 
graduated as to length. With this construction it is evident that 
the master leaves held rigidly by the band constitute a cantilever 
beam of uniform cross-section, while the remaining leaves form 
approximately a cantilever beam of uniform strength. From 
the theory of cantilever beams we find that the deflection of the 
graduated leaves for the same load and fiber stress will be 50 per 
cent, greater than that of the master leaves. Furthermore, 
when the leaves are banded together without any initial stress, 
the master leaves and the graduated leaves will deflect equal 
amounts, thus subjecting the former to a higher fiber stress. It 
is possible to make the fiber stresses in the two parts of the spring 
approximately equal by separating them by a space equal to the 
difference between the two deflections before putting the band 
in place; hence, when the band is in place and the spring is un- 
loaded an initial stress is set up in the leaves. It is customary to 
consider one of the master leaves as a part of the cantilever beam 
of uniform strength. 

114. Semi-elliptic Springs. — The following analyses and for- 
mulas pertaining to semi-elliptic springs are due to Mr. E. R. 
Morrison, who probably was the first to take into account the 
effect of the initial stress due to the band located at the mid- 
dle of elliptic and semi-elliptic springs as used in automobile 
construction. 

Let Q = total load on the spring. 

Q g = load coming upon one end of the graduated leaves. 
Q m = load coming upon one end of the master leaves. 
S g = maximum fiber stress in the graduated leaves. 
S m = maximum fiber stress in the master leaves. 
n = total number of leaves in the spring. 
n g = total number of graduated leaves. 
n m = total number of master leaves. 

(a) Initial space between leaves. — From a study of cantilever 
beams, it is evident that in order to satisfy the condition of 
equal stress in the graduated and master leaves, the following 
equation will result: 

6 LQ g 6 LQ m 



hb 2 n g . hb 2 n m 



(172) 



144 SEMI-ELLIPTIC SPRINGS [Chap. VI 

from which 

^ = ^ (173) 

n g n m 

The difference between the deflections of the graduated and 
master leaves is given by the following expression: 
6LU _ 4LU, 2LU, 

m h¥En g h¥En m h¥En m K ' 

Since — = t^-> it follows that the depth of the space which 

must be provided between the two parts of the spring before they 
are banded together is 

^-isp-STS < 175 > 

(6) Pressure due to the central band. — If the total pressure 
exerted by the central band upon the leaves is Q b , then the deflec- 
tion of the graduated leaves due to -75-, which is the pressure 
exerted by the band upon each cantilever, is as follows : 

*•' - ^ ^ 

The pressure -~ also produces a deflection in the master leaves, 
the magnitude of which is 

A' = ^!& (I7 7) 

m h¥En m { J 

Combining (176) and (177), we have 

K = l%< (178) 

Since the total deflection produced by the band is equal to the 
depth of the space provided between the two parts of the spring, 
it follows that 

* + *-;& 

from which 

A ' = -Jl^JL ^ ( i7 9) 

m dn m + 2n g nhb*E { } 

Combining (177) and (179), we get the following expression for 
the magnitude of the pressure exerted by the band: 

Q b = - "'"^ . (180) 

n(3n m +2n g ) 



Art. 115] MATERIALS FOR SPRINGS 145 

The expression for Q b just derived may be simplified by letting 
n m = kit. Since n = n g + n m , it follows that n g = n(l — k). 
Substituting these values of n g and n m in (180), we get 

Q " - 2 + fc (181) 

(c) Deflection of spring due to Q. — The deflection A of the 
spring due to Q is determined by taking the difference between 
the total deflection of the graduated leaves and that due to the 
band as given by (178); whence 

6L 3 Q g Sn m L 3 Q 



A' 



hb 3 En g ?>n m + 2n g nhb 3 E 
6 r L 3 Q -i 



k + 2 lnhb 3 E_ 
■hb 2 £ 



(182) 



nhb 2 S 
Now since Q ' = 2 (Q ff + Q m ) = Q r » we g e ^ finally that the 



deflection A due to the load Q is 



A = 



k+ 2 



m 



In the above discussion, the effect of friction between the leaves 
was not considered. 

(d) Full elliptic springs. — The anatysis given for the semi- 
elliptic springs also applies to the full elliptic type, except that 
the total deflection A will be double that of a semi-elliptic spring. 

115. Materials for Springs. — The majority of springs in com- 
mon use are made from a high-grade steel, though frequently 
brass and phosphor bronze are found more desirable. In Chapter 
II are given the specifications of several grades of steel that are 
well-adapted for the making of springs. The permissible fiber 
stress varies with the thickness or diameter of the material used 
in the construction of the spring, being higher for the smaller 
thicknesses and diameters than for the larger. According to 
Kimball and Barr's Machine Design, the maximum allowable 
stress used by an Eastern railway company in the design of steel 
leaf springs may be determined from the following formula: 

S = 60,000 + 7 -^> (184) 

in which b represents the thickness of the leaves. 



146 MATERIALS FOR SPRINGS [Chap. VI 

Quoting again from Kimball and Barr, the following formula, 
based upon an experimental investigation of springs made in the 
Sibley College Laboratories, may be used for arriving at the 
probable working stress for round stock, such as is used in 
the construction of helical springs: 

S 3 = 40,000 + 1M™, (185) 

in which d represents the diameter of the stock. 

The coefficient of elasticity E for all steels may be assumed as 
30,000,000, while that for torsion or E s may be taken at 13,000- 
000. 

The allowable working stresses and coefficients of elasticity 
for phosphor bronze and high brass spring stock are not well- 
established, and in the absence of definite knowledge relating to 
the physical constants of these materials, the following values 
obtained from various sources may be used : 

For phosphor bronze, S s varies from 20,000 to 30,000 pounds 
per square inch. 

For high brass, S s varies from 10,000 to 20,000 pounds per 
square inch. 

For high brass and phosphor bronze E = 14,000,000. 

For high brass and phosphor bronze E s = 6,000,000. 

In general, when springs are subjected to vibrations or heavy 
shock, the stresses given above for the various materials must be 
decreased from 15 to 25 per cent. 

References 

Elasticitat und Festigkeit, by C. Bach. 

Elements of Machine Design, by Kimball and Barr. 

The Strength of Materials, by E. S. Andrews. 

Elements of Machine Design, by W. C. Unwin. 

Mechanical Engineers' Handbook, by L. S. Marks, Ed. in Chief. 

Spring Engineering, by E. B. Morrison. 

Mechanical Engineers' Bocket-Book, by W. Kent. 

Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 



CHAPTER VII 

BELTING AND PULLEYS 

BELTING 

The transmission of power by means of belting may be ac- 
complished satisfactorily and efficiently when the distances 
between the pulleys are not too great. When the power to be 
transmitted is not large, round or V-shaped belts are used, the 
latter form also being used for drives with short centers. The 
materials used in the construction of belting are leather, rubber, 
cotton, and steel. 

116. Leather Belting. — The highest grade of leather belting 
is obtained from the central portion of the hide. This central 
area is cut into strips which are cemented, sewed, or riveted to- 
gether to form the desired thickness and width of belt. The 
thicknesses vary from a single hide thickness to that of four, 
the former being known as a single leather belt and the latter 
as a quadruple belt. The terms double and triple belt are used 
when two or three thicknesses are employed in the construction. 
The hides from which leather belts are made may be tanned by 
different processes. For ordinary indoor installations, the regular 
oak-tanned leather belting is well-adapted. For service in which 
the belt is exposed to steam, oil or water, a special chrome-tanned 
leather is recommended. This special tanning process is more 
or less secret and is guarded by patents. The users of this 
process claim that a more durable leather is produced, due to 
the fact the fibrous structure of the hide is preserved and not 
weakened as may result in the oak-tanning process. Leather 
belting weighs on an average about 0.035 pounds per cubic 
inch. 

(a) Commercial sizes. — Leather belting is made in the follow- 
ing widths: 

From one-half to one inch, the widths advance by 3^-inch 
increments. 

147 



148 RUBBER BELTING [Chap. VII 

From one to four inches, the widths advance by 3^-inch 
increments. 

From four to seven inches, the widths advance by 3^-inch 
increments. 

From seven to thirty inches, the widths advance by 1-inch 
increments. 

From thirty to fifty-six inches, the widths advance by 2-inch 
increments. 

From fifty-six to eighty-four inches, the widths advance by 
4-inch increments. 

The thickness of a single belt varies from 0.16 to 0.25 inch, 
while that of a double belt runs from 0.3 to 0.4 inch. 

(b) Strength of leather belting. — The ultimate strength of oak 
tanned leather runs from 3,000 to 6,000 pounds per square inch, 
the former figure applying to the lower grades of leather and 
the latter to the high-grade product. According to tests made 
on chrome-tanned leather, the ultimate strength varies from 
7,500 to 12,000 pounds per square inch. Table 35 contains infor- 
mation pertaining to the strength of leather belting, as given by 
Mr. C. J. Morrison, page 573 of The Engineering Magazine, 
July, 1916. 

117. Rubber Belting. — Rubber belting is made by fastening 
together several layers of woven duck into which is forced a 
rubber composition which subsequently is vulcanized. Belting 
of this description is used to some extent in damp places, as for 
example in paper mills and saw mills. 

A material resembling rubber, known as balata, is now used 
extensively in the manufacture of an acid- and water-proof belt. 
Balata is made from the sap of the boela tree found in Venezuela 
and Guiana. It does not oxidize or deteriorate as does rubber. 
The body of the belt, consisting of a heavy woven duck, is im- 
pregnated and covered with the balata gum, producing a belting 
material which is acid- and water-proof, and according to tests 
is about twice as strong as good leather. It is claimed that the 
heating of the belt due to excessive slippage softens the balata 
and thereby increases its adhesive properties. Due to this fact, 
it appears that balata belting is unsuitable for installations where 
temperatures of over 100°F. prevail. 

The weight of rubber belting is about 0.045 pound per cubic 
inch. 



Art. 117] 



LEATHER BELTING DATA 



149 





Table 35. — Results of 


Test on Leather Belting 




Sample 


Belt 


Breaking 
strength 


Ultimate 
strength 


Stretch in 2 inches 




Type 


Size 


Actual 


Per Cent. 


A 


1 

2 
3 
4 
5 
6 


Double 
Belt 


2X0.406 

2X0.375 
2X0.344 
2X0.3125 


4,000 
3,800 
3,200 
3,430 
3,240 
3,240 


4,930 
4,680 
3,940 
4,575 
4,700 
5,190 


0.25 
0.23 

0.27 
0.25 
0.22 


12.5 
11.5 

13.5 
12.5 
11.0 


7 

8 

9 

10 

11 

12 


Single 
Belt 


2X0.266 

2X0.25 

2X0.219 

2X0.1875 


2,230 
1,880 
2,240 
2,210 
1,840 
2,440 


4,200 
3,540 
4,226 
4,420 
4,200 
6,500 


0.23 11.5 
0.21 10.5 
0.07 3.5 
0.25 12.5 
0.23 11.5 
Too small to measure 


B 


1 
2 
3 
4 


Double 
Belt 


2X0.344 | 
2X0.281 j 


2,280 
2,460 
2,300 
2,310 


3,320 
3,580 
4,100 
4,120 


0.17 
0.27 
0.26 
0.24 


8.5 
13.5 
13.0 
12.0 


5 
6 

7 
8 


Single 
Belt 


2X0.219 
2X0.172 
2X0.1875 
2X0.172 


2,880 
1,700 
1,500 
2,180 


6,550 
4,980 
4,000 
6,380 


Too small 
0.20 
0.25 
0.18 


to measure 
10.0 
12.5 
9.0 




1 


Triple 


2X0.50 


4,510 


4,510 


0.45 


22.5 




2 
3 


Double 
Belt 


2X0.4375 
2X0.375 


4,070 
3,010 


4,650 
4,020 


0.30 


15.0 


c 






4 
5 
6 


Single 
Belt 


f 
2X0.250 J 


2,000 

850 

2,750 


4,000 
1,700 
5,500 


0.25 
0.15 


12.5 

7.5 


D 


1 
2 


Double 
Belt 


2.5X0.344 
2.5X0.3125 


3,920 
3,740 


4,558 
4,800 


0.30 
0.24 


15.0 
12.0 


E 


1 
2 
3 
4 


Double 
Belt 


2X0.344 j 

2X0.50 
2X0.375 


2,730 
2,810 
2,600 
3,240 


3,970 
4,090 
2,600 
4,300 


0.23 
0.20 
0.21 


11.5 
10.0 
10.5 




5 
6 

7 


Single 
Belt 


2X0.188 


2,010 

920 

1,420 


5,360 
2,450 
3,790 


0.20 
0.27 
0.30 


10.0 
13.5 
15.0 



150 TEXTILE BELTING [[Chap. VII 

(a) Commercial sizes. — According to one large rubber-belt 
manufacturer, the standard widths run from 1 to 60 inches as 
follows : 

From one inch to two inches, the widths advance by 3^-inch 
increments. 

From two inches to five inches, the widths advance by 3^-inch 
increments. 

From five inches to sixteen inches, the widths advance by 
1-inch increments. 

From sixteen inches to sixty inches, the widths advance by 
2-inch increments. 

The standard thicknesses run from two to eight plies. 

(b) Strength of rubber belting. — Practically no experimental 
information is available on the strength of rubber belting, though 
it is claimed by the manufacturers that a three-ply rubber belt 
is as strong as a good single-thickness leather belt. According 
to information obtained from the catalog of The Diamond Rubber 
Co., the following values may be used as representing the net 
driving tensions per inch of width for a rubber belt having an arc 
of contact of 180 degrees. 

For a three-ply belt use 40 pounds per inch of width. 

For a four- and five-ply belt use 50 pounds per inch of width. 

For a six-ply belt use 60 pounds per inch of width. 

For a seven-ply belt use 70 pounds per inch of width. 

For an eight-ply belt use 80 pounds per inch of width. 

For a ten-ply belt use 120 pounds per inch of width. 

118. Textile Belting. — Textile belts are made by weaving them 
in a loom or building them up of layers of canvas stitched 
together. The woven body or strips of canvas are treated with 
a filling to make them water-proof, and in some cases oil-proof. 
Generally, belts treated with a cheap filling are very stiff and 
hence do not conform to the pulley, making it more difficult 
to transmit the desired power. Textile belts are used more for 
conveyor service than for the transmission of power. 

(a) Commercial sizes. — The sizes of oiled and stitched duck 
belting are as follows : 

Four-ply is made in widths from 1 inch to 48 inches. 

Five- and six-ply are made in widths from 2 inches to 48 inches. 

Eight-ply is made in widths from 4 inches to 48 inches. 

Ten-ply is made in widths from 12 inches to 48 inches. 

From one to five inches, the widths vary by 3^-inch incre- 



Art. 119] STEEL BELTING 151 

ments; from five to sixteen, by 1-inch increments; and from six- 
teen to forty-eight, by 2-inch increments. 

White cotton belting is made in the following sizes : 

Three-ply having a width from 1^2 inches to 24 inches. 

Four-ply having a width from 2 inches to 30 inches. 

Five-ply having a width from 4 inches to 30 inches. 

Six-ply having a width from 6 inches to 30 inches. 

Eight-ply having a width from 6 inches to 30 inches. 

The widths of the cotton belting vary as follows : from one and 
one-half to six inches, by ^-inch increments; from six to twelve, 
by 1-inch increments; and from twelve to thirty, by 2-inch 
increments. 

119. Steel Belting. — The transmission of power by means of 
steel belts was first introduced in 1906 by the Eloesser Steel Belt 
Co. of Berlin, Germany, and at the present time this method of 
transmitting power is recognized by many German engineers as 
being superior to that in which leather belting or ropes are used. 

The steel belt is used in the same manner as the leather belt, 
except that it is narrow, thin and of very light weight. It is 
put on the pulley with a fairly high initial tension and hence runs 
without sag. The material used in making steel belts is a char- 
coal steel, prepared and hardened by a secret process. After 
rough rolling at a red heat, the metal band is allowed to cool and 
later is finished to exact size. The thicknesses vary from 0.2 
to 1 millimeter (0.0079 to 0.039 inch), and the widths range from 
30 to 200 millimeters (1.18 to 7.87 inches). The ultimate tensile 
strength of the finished material is approximately 190,000 pounds 
per square inch. 

The pulleys upon which these belts run are preferably flat, and 
are covered with layers of canvas and cork so as to increase the 
coefficient of friction. A crowned pulley may be used, provided 
the crown does not exceed approximately 33 ten-thousandths of 
the width of the belt. Steel belts are not adapted to tight and 
loose pulleys, but crossed belts will work satisfactorily, provided 
the distance between the shafts is about seventy times the width 
of the belt. 

In case the power transmitted is large, so that a single belt of 
sufficient width to give the required cross-sectional area cannot 
be obtained, two or more belts are run side by side. In putting 
steel belts on pulleys, a special clamp is used in order to measure 



152 STEEL BELTING [Chap. VII 

correctly the initial tension and at the same time to facilitate 
fitting the special plates necessary to make the joint. The de- 
sign of a proper fastening for steel belts presented a difficult 
problem, but after considerable experimental work D. Eloesser, 
now head of the firm that bears his name, perfected a joint that 
has proven very satisfactory. His first design was made of one 
piece and the ends of the belt had to be soldered in place at the 
installation. The latest design, shown in . Fig. 57, consists of 
several parts fastened together by screws e that are removable. 
The ends of the steel band are soldered to the main parts of the 
joint and the small screws/ and g passing through the triangular- 
shaped steel pieces c and d give added strength to the fastening. 
The plates a and b that form the main parts of the joint are 
curved, the curvature depending upon the size of the pulley 
upon which the belt is to run. 

(6) Experimental conclusions. — The following conclusions were 
derived from a study of a large number of tests on steel belts 
made in actual service. 




Fig. 57. 

1. Steel belts do not stretch after being placed on the pulleys, 
hence there is no necessity for taking up slack. 

2. Steel belts are not affected by variations in temperature and 
may be used satisfactorily in damp places. 

3. Steel belts will transmit the same horse power as leather 
belts having a width two to four times as great. 

4. Due to the decrease in width over leather belts transmitting 
the same power, narrower-face pulleys may be used, thus effect- 
ing a considerable saving in the cost of the pulley and in space 
due to a reduction in the general dimensions of machinery. 

5. It is claimed that the first cost of steel belting is less than 
that of leather or rubber belting. 

6. Steel belts are more sensitive and hence the pulleys, as well 
as the shafting, require more accurate alignment. 

7. Speeds as high as 19,500 feet per minute have been attained, 
and the slip at this speed was only 0.15 of 1 per cent. 



Art. 119] STEEL BELTING 153 

8. Due to the small slip, steel belts transmit power virtually 
without loss. 

9. Steel belts do not wear, and, if properly installed, are said 
to have a useful life exceeding five years. 

10. As the tension in steel belts is only a fraction, about 
one-tenth, of that used in a leather belt of the same capacity, the 
pressures on the bearings are less, thus reducing the frictional 



11. Steel belts weigh much less than leather belts of equal 
capacity, and hence reduce the frictional losses still more. 

12. Due to the extreme thinness of steel belts and the high 
speeds used, they might prove dangerous if the drive is not en- 
closed by proper guards. 

(c) Results of tests. — The following results, collected from the 
various reports recorded in several German technical journals, 
are given to show what actually has been accomplished in the 
transmission of power by means of steel belting. 

1. Under ordinary running conditions, a 4-inch steel belt is 
equivalent to an 18-inch leather belt or six manila ropes 1% inches 
in diameter. 

2. In a particular installation, a 4-inch steel belt transmitted 
250 horse power, having replaced a 24-inch leather belt. 

3. Two steel belts each 5.9 inches wide were used to transmit 
450 horse power, which formerly required 12 cables. 

4. A 6-inch steel belt 0.024 inch thick is capable of transmitting 
200 horse power, and with two such belts placed side by side on 
the same pulley, 440 horse power has been transmitted. 

5. Three 4^-inch steel belts were used to transmit 1300 horse 
power at 500 revolutions per minute of the driven pulley. The 
distance between the 122-inch driving and 63-inch driven pulleys 
was 46 feet. 

6. In another installation, 75 horse power was transmitted by 
a 6-inch steel belt running over pulleys 108 and 51 inches in 
diameter, located on 76-inch centers. 

(d) American experiments on steel belting. — In 1911 or 1912, 
the General Electric Co. made a series of experiments with steel 
belts, and came to the conclusion that they were not entirely 
satisfactory. The thicknesses of the belts used in these experi- 
ments varied from 0.007 to 0.018 inch. A %-inch belt 0.01 
inch thick was capable of transmitting 150 horse power con- 
tinuously for 17 hours at a speed of 20,000 feet per minute. This 



154 BELT FASTENERS [Chap. VII 

belt was made of cold-rolled steel and the initial tension put 
on the belt in order to give the above results was 90,000 pounds 
per square inch. The General Electric Co. found that steel belts 
will not run satisfactorily on the ordinary steel pulleys, and the 
best results were obtained with a leather-faced pulley. No doubt 
the following are some of the reasons why the results obtained 
by the General Electric Co. from their investigation on steel 
belting were not as promising as those found by the German 
engineers : 

1. Not as good a grade of steel available for making the band. 

2. Probably during the early stages of preparing the band, im- 
proper treatment gave rise to scale troubles. 

3. Difficulty in the process of annealing. 

4. Lack of time for further research work. 

120. Belt Fastenings. — Fastenings of various forms are used 
for joining the ends of a belt, but none of them is as strong and 
durable as the scarfed and glued splice, which when made care- 
fully is but little weaker than the belt proper. Of necessity, the 
scarfed and glued joint or cemented splice is adapted to installa- 
tions in which the slack of the belt is taken up by mechanical 
means, and where careful attention is given to belting by com- 
petent workmen. Probably the oldest form of fastening, as well 
as that used most commonly, is to join the ends of a belt by 
means of rawhide lacing. Not infrequently belts are laced to- 
gether with wire, and such joints run very smoothly, especially 
if made with a machine, and are considerably stronger than the 
rawhide laced joint, as is indicated in Table 36. Patented 
metal fasteners in the form of hooks, studs, and plates are also 
in use and have the advantage that they are cheap and applied 
very easily and quickly. Some of the metal fasteners are too 
dangerous to be used on belts that must be touched by hand, and 
for that reason some states have legislated against their use. 

Tests of belt joints. — Tests of various types of belt joints 
were made at the University of Wisconsin, also at the University 
of Illinois. In The Engineering Magazine of July, 1916, Mr. C. 
J. Morrison presented a valuable article entitled "Belts — Their 
Selection and Care," in which he gives considerable information 
pertaining to the strength of leather belts and the joints used 
with such belting. In Table 36 is given information pertaining 
to the strengths and efficiencies of the various types of leather 
belt joints tested by Mr. Morrison. It should be understood 



Art. 121] 



BELT TENSIONS 



155 



that the term "efficiency" in this case is used in the sense as 
when applied to riveted joints. 

Table 36. — Strength of Leather Belt Joints 



Type of joint 



Breaking 

load, 
pounds 



Efficiency, 
per cent. 



Cemented 
splice 



Cement only 

Cement and shoe pegs 

Cement and small copper rivets . 

Cement and small copper rivets . 

Cement and large copper rivets. . 

Wire, machine-laced 

Wire, hand-laced 

Rawhide with small holes 

Rawhide with large holes 

Metal hooks 

Metal studs 



2,440 
2,430 
2,170 
2,060 
2,040 
5,850 
5,330 
4,100 
3,200 
2,270 
1,950 



100.0 
99.6 
88.9 
84.4 
83.6 
90.0 
82.0 
63.0 
49.0 
35.0 
30.0 



STRESSES IN BELTING 

121. Tensions in Belts. — A belt transmits power due to its 
friction upon the face of the pulley. This transmitting capacity 
depends upon the following important factors: 

(a) The allowable net tension in the belt. 

(b) The coefficient of friction existing between the belt and 
pulley. 

(c) The speed at which the belt is running. 

Net tensions. — The net tension represents the capacity of 
the belt and depends upon the maximum allowable tension, 
the coefficient of friction, the angle of contact that the belt 
makes with the pulley, the material of both the belting and the 
pulley, the diameter of the pulley, and the velocity of the belt. 
The net tension is not a constant as is frequently assumed, but 
it varies with the speed. Let two pulleys be connected by a 
belt as shown in Fig. 58, and assume that no power is being trans- 
mitted, except that required to overcome the frictional resistance 
on the bearings due to the initial tension with which the belt was 
placed on the pulleys. Due to this initial tension, which is the 
same on both the running on and off sides of the pulleys, the belt 
exerts a pressure upon the face of the pulleys. This pressure in 
turn induces a frictional force on the rim capable of overcoming 



156 



RATIO OF BELT TENSIONS 



[Chap. VII 



an equivalent resistance, tending to produce relative motion 
between the belt and pulley. The tensions in the two parts 
of the belt will change as soon as power is transmitted, say from 
a to 6, causing that in the pulling side to increase and that in the 
running off side to decrease. Representing these tensions by the 
symbols 7\ and T 2 , we see that the force causing the driven pulley 
b to rotate is the difference of these tensions, or T x — T 2 . This 
difference is known as the net tension. 

It is evident that due to this difference in tension in the various 
sections of the belt, a unit length of the belt in running from the 
point A to B, decreases in length due to its elasticity. From 
this it follows that the driver a delivers a shorter length of belt at 




B than it receives at, A and furthermore, that the velocity of the 
pulley face and that of the belt are not equal. A similar action 
occurs on the pulley b. This action is known as belt creep and 
results in some loss of power. 

122. Relation between Tight and Loose Tensions. — The 

horse power delivered by a belt may be determined as soon as 
the net tension and the speed are established; hence it is im- 
portant to derive the relation existing between the tight and 
loose tensions. 

Let A = cross-sectional area of the belt in square inches. 
C = centrifugal force of an elementary length of belt. 

5 = allowable working stress of the belt. 
b = width of belt. 

t = thickness of belt. 

v = velocity of belt, in feet per second. 

w = weight of belt, pounds per cubic inch. 

H = coefficient of friction. 

6 = total angle of contact, expressed in radians. 



Art. 122] 



RATIO OF BELT TENSIONS 



157 



In Fig. 59 a short portion of the belt has an arc of contact sub- 
tending the angle Ad at the center of the pulley. Let the tension 
at one end be T and at the other (T + AT); evidently each of 

these tensions makes an angle ~ ^ with the vertical center 

line. The pressure between the portion of the belt and the pulley 
rim is designated by the symbol N, and the force of friction 
between them is /jlN. In addition to these forces, we have the 
centrifugal force C acting radially as shown in the figure. The 
magnitude of the centrifugal force is given by the following 
expression: 

C= 12A ^" 2 (186) 



T+AT 




Fig. 59. 



The piece of belt referred to above is held in equilibrium by 
the five forces T, (T + AT), N, ixN^nd C. The summation of 
the horizontal and vertical components, respectively, gives the 
following equations: 



AT cos ~ + fiN 







(187) 



(2T + AT)sin^- - N - C = 



(188) 



Eliminating N 



n (2 T + AT) sin ~ - AT cos -^ - mC = 



158 RATIO OF BELT TENSIONS Chap. VII] 

A/? 

Dividing through by -~-, and passing to the limit, we get 

. Ad 

qiri 

/x hm (2 7 1 + AT) lim — = 2 km — — hm cos — 

^ v M_ g Ad 2 

~2~ 

whence 

where 

12 Awv 2 

k = 

9 
Separating the variables 



M (T -k) (189) 



/:a=.j> 



Integrating, we find that the relation between the tight and loose 
tensions is as follows : 

|^| = f- U») 

■ From (190), we find that the net tension is 



T x 



- T 2 = {T, - k) [^r 1 ] (191) 



Substituting in (191) the value of 2\ in terms of b, t and S, we 
have 



T 1 



-T^u[s-^\[^] 



r 12 wv 2 ~\ re 1 * 6 — 1~| 

Denoting the terms S and — — e — by the sym- 
bols m and n, respectively, we get finally 

T 1 - T 2 = mnbt (192) 

Having determined the magnitude of the net tension from 
(192) and knowing the speed v, the horse power delivered may be 
calculated from the relation 

H^in-T,) (193) 

123. Coefficient of Friction. — There is much diversity of 
opinion regarding the working coefficient of friction, but in general 
it depends upon the material of the belt and the condition of the 



Art. 124] 



COEFFICIENT OF FRICTION 



159 



belt, the permanent slip, whether the load is steady or fluctuat- 
ing, the diameter of the pulley and the material of which it is 
made, and the speed of the belt. In view of the foregoing, the 
coefficient of friction cannot be assumed as an average for all 
speeds, as is so frequently done in belting calculations. It is 
practically impossible to derive an expression for /jl in terms of 
all of the factors mentioned above, but the following formula 
proposed by Mr. C. G. Barth has been found to give fairly satis- 



n^ 5000 














4000 




Speed 


of 


Belt 
3000 


-ft 


per 


m 


n 


2000 
















1000 


0.50 


















































































£0.45 

'+- 

ih 
040 


































































































■5 0-35 
































S- 












y 














0.30 


I 


/ 










) 






IC 









I 


)0 






3C 









4( 









5 


30 






b 


30 






7 


DO 






8 


30 






9 


30 






10 


00 



Speed of Belt -ft per min. 
Fig. 60. 



factory results in practice for leather belting on cast-iron or 
steel-rim pulleys. 

" = o:54 -50^FT' (194) 

in which V represents the velocity of the belt in feet per minute 
The Barth formula for n, as given by (194), has been evaluated 

for various values of V, and the results obtained are shown in 

graphic form in Fig. 60. 

124. Maximum Allowable Tension. — The maximum allowable 

tension that may be put upon a belt depends upon the quality of 

the material, the permanent stretch of the belt, the imperfect 



160 



SELECTION OF BELT SIZE 



[Chap. VII 



elasticity of the belting material, and the strength of the joints 
in the belts. In Table 37 are given the average values for 
the ultimate strengths of leather belting, as given by Morrison 
in his article referred to previously. To arrive at the magnitude 
of the allowable working stress S for leather, multiply the ulti- 
mate strength by the so-called efficiency of the joint and divide 
the product thus obtained by the assumed factor of safety. As 
an aid in the solution of belt problems, the several factors just 
mentioned, as well as the allowable working stresses for the im- 
portant joints used in connection with leather belting, are given 
in Table 38. 

Table 37. — Average Ultimate Strength op Leather Belting 



Mfr. 


No. of 
samples 


Best 


Poorest 


Average 


Remarks 


A 


12 


6,500 


3,549 


4,611 




B 


8 


6,550 


3,303 


4,614 




1 


6 


5,500 


1,700 


4,062 


Poorest broke in the 


° 










splice. 


I 


5 


5,500 


4,013 


4,532 


Omitting 1,700 sam- 
ple. 


D 


2 


4,800 


4,558 


4,679 




E 


7 


5,360 


2,453 


3,800 





Table 38. — Working Stresses for Leather Belting 



Type of joint 


Ultimate 
strength 


Efficiency 
of joint 


Factor of 
safety 


Working 

stress S 


Cemented 


4,300 


0.98 
0.88 
0.80 
0.60 


10 


420 


w - f Machine-laced. . . . 
\ Hand4aced 


380 
340 
260 







125. Selection of Belt Size. — Having arrived at the allowable 
working stress in a belt, and knowing the magnitude of the net 
driving tension P as well as the angle of contact d and the coef- 
ficient of friction /jl, the area of the belt may be calculated by 
means of (192). From the conditions of the problem, either 
the width of the belt or its thickness may be established; hence 
the remaining dimension may be determined. Now the selec- 
tion of the proper belt thickness is, in general, determined by the 
diameter of the smallest pulley used in the transmission. If the 



Art. 126] TAYLOR'S EXPERIMENTS 161 

belt is thick relative to the diameter of the smallest" pulley, the 
result will be an unsatisfactory drive, due to the excessive slip- 
page and belt wear, as well as the excessive loss of power. In 
addition to the points just mentioned, the result of running a 
thick belt over a small pulley will be a considerable decrease in 
the life of the belt. 

Satisfactory belt service, as well as long life, is secured if the 
diameter of the smallest pulley in the transmission is made not less 
than 12 inches if a double belt of medium or heavy weight is 
used ; for a triple belt, the minimum diameter of pulley should be 
20 inches, and for a quadruple belt, 30 inches. The selection of a 
belt thickness may also be influenced to a certain degree by the 
fact that good reliable single belts are hard to obtain in widths 
exceeding 12 to 15 inches. A rule occasionally used for the limit- 
ing size of a single belt is as follows: "A single belt should never 
be used where the width is more than four-thirds the diameter of 
the smallest pulley." 

126. Taylor's Experiments on Belting. — In volume XV of the 
Transactions of the American Society of Mechanical Engineers, 
Mr. F. W. Taylor reports "A Nine Years' Experiment on Belt- 
ing" carried on at the Midvale Steel Co. This paper gives some 
valuable data on the actual performance of belts, and a satis- 
factory abstract of it is impossible in this chapter. The conclu- 
sions, thirty-six in number, given in the paper are based upon the 
cost of maintaining the belts in good condition, including time 
lost in making repairs, as well as other considerations. The 
following are some of the conclusions : 

(a) Thick narrow belts are more economical than thin wide 
ones. 

(6) The net driving tension of a double belt should not exceed 
35 pounds per inch of width, but the initial tension may be double 
that value. 

(c) The most economical belt speed ranges from 4,000 to 
4,500 feet per minute. 

(d) For pulleys 12 inches in diameter or larger double belts 
are recommended. 

For pulleys 20 inches in diameter or larger triple belts are 
recommended. 

For pulleys 30 inches in diameter or larger quadruple belts 
are recommended. 



162 TANDEM-BELT TRANSMISSION [Chap. VII 

(e) The joints should be spliced and cemented rather than 
laced with rawhide or wire, or joined by studs or hooks. 

(/) Belts should be cleaned and greased every five or six 
months. 

(g) The best distance between centers of shafts is from twenty 
to twenty-five feet. 

(h) The face of a pulley should be 25 per cent, wider than the 
belt. 

127. Tandem-belt Transmission. — Not infrequently two belts, 
one placed on top of the other, are used to transmit power from 
one pulley to two separate pulleys. This arrangement is known 
as a tandem-belt drive. The outside belt travels at a somewhat 
higher speed than the inner, and this fact must not be lost sight 
of when a tandem-belt transmission is being designed in which 
the speeds of the two driven pulleys must be the same. Experi- 
ence with tandem-belt drives has shown that the best results 
are obtained when both belts are of the same thickness, prefer- 
ably of double thickness, and are placed upon the pulleys with 
the same initial tension. Due to the higher coefficient of fric- 
tion between leather and leather, practically all the slip will 
occur between the pulley and the inner belt. To arrive at the 
proper size of a belt required for a tandem drive, proportion each 
belt according to the power it must transmit. 

128. Tension Pulleys. — Whenever possible, it is well to provide 
means of releasing the initial tension in belts during extended 
periods of idleness. In some case's, as in electrical machinery, 
this is accomplished by mounting the machines on rails, thus 
providing means for changing the distance between the centers 
of the pulleys. To a certain extent, the practice of making the 
loose pulley on machine drives smaller in diameter, will relieve 
the best tensions. There are, however, many belting installations 
where neither of these methods could be used, and in many of 
these cases tension pulleys designed and installed properly will 
improve the transmission. 

Lenix system. — In the Lenix system, the tension pulley is 
placed on the slack side of the belt as near to the smaller pulley 
in the transmission as is practicable. The general features of 
this system are shown in the two radically different installations 
represented in Figs. 61 and 62. The tension pulley is carried on 
an arm pivoted on the axis of the small driving pulley, and by 



Art. 128] 



TENSION PULLEYS 



163 



means of a weight the required tension may be put on the slack 
belt. In the installation shown in Fig. 61, the tension on the 
belt is changed by increasing or decreasing the leverage of the 




Fig. 61. 



tension weight. It is evident from an inspection of Figs. 61 and 
62, that a large arc of contact is obtained by means of this system 
and for that reason the tension in the belt may be reduced. 




The diameter of the tension pulley should never be made less 
than that of the smallest pulley in the drive. The only losses 



164 



PULLEYS 



[Chap. VII 



chargeable to the tension pulley are those due to journal friction, 
which, if the apparatus is properly designed and erected, are 
small and have practically no effect on the efficiency of the trans- 
mission. Some additional advantages of tension pulleys are as 
follows : (1) the initial tension of the belt may be regulated very 
accurately and may be maintained at the proper magnitude; (2) 
during periods when the drive is not in use the belt may be re- 
lieved of the initial tensions. 



PULLEYS 

129. Types of Pulleys. — (a) Cast-iron pulleys. — Pulleys are 
made from various kinds of materials, cast iron, however, being 
the most common. As far as the cost of manufacture is con- 
cerned, cast iron is ideal since it can be cast in any desired shape, 
though precautions must be taken in the foundry when light- 







Fig. 63. 



weight pulleys are cast. If the metal in the various parts of the 
pulley is not distributed correctly, shrinkage stresses due to 
irregular cooling are likely to reduce the useful strength of the 
material. To partly overcome this trouble, pulleys are split in 
halves. Careless moulding in the foundry generally produces 
pulleys having rims that are not uniform in thickness, thus caus- 
ing them to run out of balance. This defect is rather serious in 
a high-speed transmission, though the pulley can be balanced by 
attaching weights at the lightest points. The centrifugal force 



Art. 129] 



PULLEYS 



165 



due to these weights will set up severe stresses in the weak rim 
and may cause it to burst. 

(b) Steel pulleys. — A type of pulley introduced to overcome 
some of the defects of cast-iron pulleys consists of a cast-iron 
hub and arms to which is riveted a steel rim. Pulleys built in 
this way are lighter than cast-iron ones for the same duty, but 
trouble may result with the fastenings as they may work loose 
due to the heavy loads transmitted. Pulleys built entirely of 
steel are also used, and are looked upon with favor by many 
engineers. In Figs. 63 and 64 are shown the designs of a small 
and large pulley as manufactured by The American Pulley Co. 
of Philadelphia. An inspection of Figs. 63 and 64 shows that 
the construction adopted for these pulleys gives a maximum 
strength for a minimum weight, and furthermore, the windage 
effect at high speeds is small. 




(c) Wood pulleys. — Wood pulleys in the smaller sizes generally 
consist of a cast-iron hub upon which is fastened a wood rim built 
up of segments of well-seasoned maple. In the larger sizes, they 
are always made in the split form and are built entirely of wood. 
Due to atmospheric conditions, wood pulleys are very likely to 
warp or distort, which may cause trouble at high speeds. 

(d) Paper pulleys. — Pulleys made of paper are also in com- 
mon use. As shown iD Fig. 65, such a pulley consists of a web 
and rim built up of thin sheets of straw fiber cemented together 
and compressed under hydraulic pressure. To secure additional 
strength in the rims, wooden dowel pins extend through the rim 
and web as shown in the figure. The webs are clamped securely 
between the flanges of the cast-iron hub as shown. 

(e) Cork insert pulleys. — Frequently pulleys are lagged with 



166 



TRANSMITTING CAPACITY OF PULLEYS [Chap. VII 



leather or cotton belting in order to increase the coefficient of 
friction between the belt and pulley. However, such lagging 
wears out quickly and must be renewed, thus increasing materi- 
ally the cost of upkeep of the transmission. It has been found 
by an extended series of experiments, conducted by Prof. W. M. 
Sawdon of Cornell University, that the transmitting capacity of 
practically any type of pulley can be increased by fitting cork 
inserts into the face. The corks are pressed into the face and 
allowed to protrude above the surface of the material of the face 
not to exceed 3^2 inch. These cork inserts dc not wear down 
nearly as rapidly as the lagging; however, the first cost is con- 
siderably more. 





Fig. 65. 



130. Transmitting Capacity of Pulleys. — In September, 1911, 
before the National Association of Cotton Manufacturers, Prof. 
W. M. Sawdon read a paper entitled "Tests of the Transmitting 
Capacities of Different Pulleys in Leather Belt Drives," in which 
he presented the results of an extended investigation on the trans- 

Table 39. — Comparative Transmitting Capacities of Pulleys 



Type of pulley 



Relative capacities at various 
slips 



1 

per cent. 



1H 2 

per cent, per cent. 



1 Cast iron 

2 Cast iron with corks projecting 0.04 inch. 

3 Cast iron with corks projecting 0.015 inch 

4 Wood 

5 Wood with corks projecting 0.075 inch... 

6 Wood with corks projecting 0.03 inch 

7 Paper 

8 Paper with corks projecting 0.087 inch 

9 Paper with corks projecting 0.015 inch... 



100.0 
133.5 
139.3 
136.3 
130.7 
130.7 
160.7 
149.0 
150.2 



100.0 
119.7 
124.1 
118.8 
116.8 
118.2 
151.7 
135.5 
145.3 



100.0 
107.0 
112.0 
105.6 
104.8 
104.8 
137.3 
122.0 
133.0 



Art. 131] 



PROPORTIONS OF PULLEYS 



167 



mitting capacities of pulleys. In this paper, Prof. Sawdori' gave 
a table of relative capacities based on the same arc of contact 
and the same belt tensions, which may prove useful in the solution 
of belt problems. The data given in Table 39 were derived from 
this paper. In using the table it should be kept in mind that 
the figures are relative and, strictly speaking, apply only to the 
conditions of operation prevailing during the tests. However, 
the results may be used tentatively until further data per- 
taining to this subject are available. 

131. Proportions of Pulleys. — (a) Arms. — It is very seldom 
that a designer is called upon to design cast-iron pulleys except 




Fig. 66. 



for an occasional special purpose, and for that reason it is best to 
leave the general design of standard pulleys to the pulley manu- 



Table 40. — Proportions of 


Extra-heavy Cast 


-iron Pulleys 








Dimensions 






TV 














l 


2 


3 


4 


5 


6 


12 


0.38 


% 


% 


1%6 


1M 


4 


15 


0.40 


% 


1^2 


1% 


1% 


4M 


18 


0.42 


21 /Z2 


lHe 


1% 


1% 


4K 


24 


0.46 


1H2 


1M 


2 


2^6 


4% 


30 


0.50 


1«2 


IKe 


2% 


2% 


7 


36 


0.54 


IK 


1% 


2%6 


3M 


7 


42 


0.58 


1%6 


1% 


sy 8 


3% 


8 


48 


0.62 


1% 


1^6 


3% 


4% 


8 


• 54 


0.66 


IV2 


2 


4^ 


5Ke 


9K 


60 


0.70 


1 1 %2 


2Ke 


4K 


5M 


9K 



168 PROPORTIONS OF PULLEYS [Chap. VII 

facturer. In Fig. 66 is represented an ordinary cast-iron pulley, 
and the proportions of various sizes of extra-heavy double-belt 
pulleys given in Table 40 may serve as a guide in the design of 
special pulleys. 

A series of tests made on various kinds of pulleys by Prof. 
C. H. Benjamin, the results of which were published in the 
American Machinist of Sept. 22, 1898, proved rather conclusively 
that the rim of a pulley does not distribute the torsional moment 
equally over the arms as is so frequently assumed. In every 
test made, the two arms nearest the tight side of the belt gave 
way first and in almost all cases rupture of the arm occurred at 
the hub. As a result of these tests, Prof. Benjamin suggests that 
the hub end of the arm should be made strong enough so that it 
is capable of resisting a bending moment equivalent to 

M = ^{T l -T 2 ), (195) 

in which 

D = diameter of the pulley in inches. 
n = the number of arms. 

This means that one-half of the arms are considered as effect- 
ive. The dimensions of the arm at the rim should be made such 
that the sectional modulus is only one-half of that at the hub. 

The various manufacturers differ as to the number of arms to 
be used with the different sizes of pulleys, but the following 
suggestions may be found useful : 

Use webs for pulleys having a diameter of 6 inches or less. 

Use 4 arms for pulleys having a diameter ranging from 7 to 
18 inches. 

Use 6 arms for pulleys having a diameter ranging from 18 to 
60 inches. 

Use 8 arms for pulleys having a diameter ranging from 60 to 
96 inches. 

When the face of a pulley is wide, a double set of arms should 
always be provided. 

The working stress to be used in calculating the dimensions of 
the arms by means of (195) varies within very wide limits. An 
investigation of the arms of pulleys having a diameter of from 
12 to 96 inches and a face of 4 to 12 inches gave stresses varying 
from 200 to 1,500 pounds per square inch. The latter stress is 
obtained in the smaller pulleys and the former with the larger 
diameters. 



Art. 132] TIGHT AND LOOSE PULLEYS 169 

(b) Rim. — According to Mr. C. G. Barth, the face of the 
pulley should be considerably wider than the belt that is to run 
on it, and in order to establish uniform proportions, he proposed 
the following formulas : 

/= 1^6 + | inch. (196) 

/=1 l 6 + ^ inch - (197) 

Formula (196) is the one that should be used wherever possible, 
but occasionally due to certain restrictions as to available space, 
(197) may have to be used. In connection with these formulas, 
Mr. Barth recommends that the height of the crown should be 
determined by the formula 

c=-g (198) 

For proportions of the thickness of the rim, the data given in 
Table 40 may be of service. 

132. Tight and Loose Pulleys. — In his consulting work, Mr. 
Barth has found the need of well-designed tight and loose pulleys. 
After a thorough study of the conditions under which such pul- 
leys must operate, he developed the design shown in Fig. 67. 
Furthermore, he standardized the design, and the formulas be- 
low give well-proportioned sleeves and pulleys for shaft diameters 
from \]/2, to 4 inches, inclusive. The face and height of crown 
for these pulleys are based on formulas (196) to (198) inclusive. 
The formulas giving the proportions of the pulley hub and sleeve 
a are based on the diameter d of the shaft. 

d\ = 1.5 d + 1-5 inches 

dz = 1.5 d + 1 inch 

d z = 1.375 d + 0.75 inch 

di = 1.25 d + 0.25 inch (199) 

e = ^ + 0.125 inch 
16 

m = - + 0.75 inch 
6 

The formulas listed below give proportions of the loose pulley 
rim, and are based upon the width of the belt running on the 
pulleys. The belt width as given by Mr. Barth varied from 2 
to 6 inches, inclusive. 



170 



TIGHT AND LOOSE PULLEYS [Chap. VII 



/-!&» + fin* 

,--A + l|moh 

h =f + 2g 

&= 16 + 16 mch 



(200) 





Art. 133] 



V BELTING 



171 



The common tight and loose pulleys that are used in the 
majority of installations differ considerably from the design 
discussed above in that both pulleys are generally made alike, 
and in many cases neither pulley is crowned. 



V BELTING 

133. Types of V Belts. — As stated in the first part of this 
chapter, V belts are used when it is desired to transmit light 
power; for example, in driving the cooling fan and generator on 



\ ..r*- -*'- / X -i. .1^ S \_jci 



F5 



r;~? 



(a) 



b d 




(b) 




automobiles, and transmission drives on motorcycles. It is 
also used for belting electric motors to pumps and ventilating 
fans, when the distances between the shafts are short. Several 
forms of V belting are shown in Fig. 68. 

(a) Block type. — The construction used in the block type of 
V belt is shown in Fig. 68(a). It consists of a plain high-grade 
and very pliable leather belt to which are cemented and riveted 



172 V BELTING [Chap. VII 

equally spaced V blocks, also made of leather. For light loads, 
a single belt is used; and for heavy service, a wide belt is fitted 
with several rows of V blocks. The angle adopted in this design 
is 28 degrees, and according to the manufacturers of this belt, 
the maximum speed should not exceed 3,000 feet per minute. 
The belt shown in Fig. 68(a) is also used successfully on high 
pulley ratios, though the best results are obtained if the ratio 
does not exceed 6 or 7 to 1. In addition to giving good service 
on high-ratio pulleys, the block type of V belt also works suc- 
cessfully on pulleys located close together. The following recom- 
mendations were furnished by the Graton and Knight Mfg. 
Co.: 

1. For a 2, 3, or 4 to 1 ratio, the minimum center distance 
equals the diameter of the larger pulley plus twice the diameter 
of the smaller one. 

2. For a 5, 6, or 7 to 1 ratio, the minimum center distance 
equals the diameter of the larger pulley plus three times the 
diameter of the smaller one. 

3. For a 8, 9, or 10 to 1 ratio, the minimum center distance 
equals the diameter of the larger pulley plus four times the 
diameter of the smaller. 

(b) Chain type. — The construction shown in Fig. 68(o) is of 
the chain type, and consists of double links made of oak-tanned 
sole leather connected together by central links c made of steel. 
The steel links are fitted with short pins d to which the leather 
links are attached. To add strength to the belt as well as to 
afford a fair bearing for the pins d, vulcanized fiber links b are 
used between the leather and steel links. An ordinary wood 
screw clamps the two sets of double links together, as illustrated 
in the figure. All the driving is done by the leather links, and 
the angle used is 28 degrees. 

Another construction of the chain type V belt made entirely 
of steel, except the part coming into contact with the pulley, is 
shown in Fig. 68 (c) . The material used for lining the steel driving 
members is not leather but a specially treated asbestos fabric. 

134. Force Analysis of V Belting.— To determine the relation 
existing between the tight and loose tensions in a V-belt power 
transmission, we may follow the method given in Art. 122. 

Let w = weight per foot of belt. 
2 /3 = total angle of the V groove. 
C,v, n and 6 same meaning as in Art. 122. 



Art. 134] 



V BELTING 



173 



Referring to Fig. 69 and taking the summation of the horizontal 
and vertical components, respectively, of all forces acting upon a 
small portion of the belt, we get 



ATcos 



Ad 



2 M iV 



Ad 



(2 T + AT) sin -j - 2 N sin - C = 



(201) 
(202) 



The magnitude of the centrifugal force C in this case is given 
by the following equation : 

C = ^ (203) 



Eliminating N in (201) and (202) and taking the limits of the 
resultant expression, we finally get 

,lT (204) 






— y 


V A, 


k. 




m 






'M. 


k 


i 


^ 



Integrating (204) between the proper limits for T and 0, we 
obtain 



y _ "21 



= esin/3 



(205) 



The net driving tension of the belt is 



ri - r, - [r, - ^- ! ] 



ring - i 

6 sini/3 



(206) 



To determine the horse power transmitted, substitute the 
magnitude of the net driving tension obtained from (206) in 
(193). 



174 REFERENCES [Chap. VII 

References 

Die Maschinen Elemente, by C. Bach. 

Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 

Leather Belting, by R. T. Kent. 

Experiments on the Transmission of Power by Belting, Trans. A. S. M. E., 
vol. 7, p. 549. 

Belt Creep, Trans. A. S. M. E., vol. 26, p. 584. 

The Transmission of Power by Leather Belting, Trans. A. S. M. E., vol. 31, 
p. 29. 

The Effect of Relative Humidity on an Oak-tanned Leather Belt, Trans. 
A. S. M. E., vol. 37, p. 129. 

Tensile Tests of Belts and Splices, Amer. Mach., Oct. 10, 1912. 

Belt Driving, The Engineer (London), Apr. 23 and 30, 1915. 

The Design of Tandem Belt Drives, Amer. Mach., Apr. 1, 1915. 

Theory of Steel Belting, Zeitschrift des Vereins Deutscher Ingenieure, Oct. 
21, 1911. 

Transmission of Power by Means of Steel Belting, Dinglers, Sept. 2 and 9, 
1911. 

The Practicability of Steel Belting, Amer. Mach., Nov. 21, 1912. 



CHAPTER VIII 
. MANILA ROPE TRANSMISSION 

Ropes used in engineering operations are made of a fibrous 
material such as manila, hemp and cotton, or of iron and steel. 
As to the kind of service, ropes may be classed as follows: (a) 
those used for the hoisting and transporting of loads; (6) those 
used for the transmission of power. 

FIBROUS HOISTING ROPES 

135. Manila Hoisting Rope. — Manila rope is manufactured 
from the fiber of the abaca plant, which is found only in the 
Philippine Islands. It has a very high tensile strength, tests 
made at the Watertown Arsenal showing that it exceeds 50,000 
pounds per square inch. In making the rope, the fibers are 
twisted right-handed into yarns; these yarns are then twisted 
in the opposite direction forming the strands, and to form the 
finished rope a number of strands are twisted together, again in 
the right-hand direction. 

Practically all manila rope used for hoisting purposes has four 
strands except the sizes below % inch, which are made with three 
strands. For drum hoists using manila ropes, the maximum 
speed attained under load seldom exceeds 1,000 feet per minute, 
generally being nearer 300 feet per minute. The permissible 
working loads of the various sizes of manila ropes used for hoist- 
ing service are given in Table 41. 

136. Sheave Diameters. — A rope in passing over sheaves is 
subjected to a considerable amount of internal wear, due to the 
fibers sliding upon each other. The smaller the diameter of the 
sheave the greater this sliding action becomes ; hence to decrease 
the wear, large sheaves should be used. In addition to the 
internal wear there is also wear on the outside of the rope due 
to the friction between it and the sides of the grooves of the 
sheave. It is evident, therefore, that the grooves should be 
finished very smooth. Again, the arrangement of the various 
elements that make up the hoisting apparatus may be such that 
an excessive number of bends is introduced, thus increasing the 
wear. 

175 



176 



MANILA ROPE 
Table 41. — Manila Rope 



[Chap. VIII 





For hoisting 


For Transmission 


Diameter 
















inches 


Weight 


Ultimate 


Mini- 
mum 


Weight 


Ultimate 


Maximum 


Mini- 
mum 




per 
foot 


strength 


sheave 
diam. 


per 
foot 


strength 


allowable 
tension 


sheave 
diam. 


H 


0.018 


620 












Vl6 


0.024 


1,000 












% 


0.037 


1,275 












Ke 


0.055 


1,875 












y% 


0.075 


2,400 












He 


0.104 


3,300 












Vs 


0.133 


4,000 












Va 


0.16 


4,700 




0.21 


3,950 


112 


28 


% 


0.23 


6,500 




0.27 


5,400 


153 


32 


1 


0.27 


7,500 


8 


0.36 


7,000 


200 


36 


1H 


0.36 


10,500 


9 


0.45 


8,900 


253 


40 


m 


0.42 


12,500 


10 


0.56 


10,900 


312 


46 


m 


0.55 


15,400 


11 


0.68 


13,200 


378 


50 


m 


0.61 


17,000 


12 


0.80 


15,700 


450 


54 


m 


0.75 


20,000 


13 


0.92 


18,500 


528 


60 


m 


0.93 


25,000 


14 


1.08 


21,400 


612 


64 


2 


1.09 


30,000 




1.40 


28,000 


800 


72 


2M 


1.5 


37,000 




1.80 


35,400 


1,012 


82 


2M 


1.71 


43,000 




2.20 


43,700 


1,250 


90 



Experience has shown that manila ropes give good service and 
will last a reasonable length of time in hoisting operations when 
the sheaves for the various sizes of ropes are made according to 
the diameters given in Table 41. 

137. Stresses in Hoisting Ropes. — In hoisting operations ropes 
are wound upon drums, and sheaves are used for changing the 
direction of the rope. In passing over sheaves or onto drums, 
the rigidity of the rope offers a resistance to bending which must 
be overcome by the effort applied to the pulling side of the rope. 
To determine the relation that exists between the effort P and 
the resistance Q for a rope running over a guide sheave, the 
following method may be used : 

Let D = pitch diameter of the sheave. 
d = diameter of the sheave pin. 
yu = coefficient of journal friction. 
t) = efficiency. 



Art. 137] 



STRESSES IN HOISTING ROPE 



177 



On the rimning-on side of the sheave shown in Fig. 70, the 
outer fibers, due to the bending of the rope, are in tension while 
the inner fibers are in compression. These tensile and com- 
pressive stresses when combined with the tension distributed 
uniformly over the section will produce a resultant which has its 




point of application to the left of the center line of the rope, 
a distance designated by the symbol s. The resultant must be 
equal to Q, from which it follows that rope stiffness may be con- 
sidered as having the same effect as increasing the lever arm of 
the resistance Q. 

By applying the same line of reasoning to the running-off side, 
it may be shown that the rigidity of the rope has the effect of 
decreasing the lever arm of the effort P by an amount which may 
be taken as approximately equal to s. Introducing friction at 
the sheave pin and taking moments about the line of action of 
the resultant pressure upon this pin, we obtain 



*-[&£if2«-<* 



(207) 



Since the efficiency of a mechanism is defined as the ratio of 
the useful work done to the total work put in, it i& evident that 
in the case of the ordinary rope guide sheave 



Q 



(208) 



178 



BLOCK AND TACKLE 



[Chap. VIII 



m 



138. Analysis of Hoisting Tackle. — Analyses of systems of 
hoisting tackle or so-called pulley blocks are readily made with 
the aid of the principle discussed in the preceding article. The 

application of this prin- 



ciple will be shown by 
an example. 

Common block and 
tackle. — The common 
• block and tackle con- 
sists of two pulley 
blocks, each block hav- 
ing a series of sheaves 
mounted side by side on 
the same axle or pin. 
The number of sheaves 
varies in ordinary hoist- 
ing operations from two 



Fig. 71. 



to four, but when used in connection with wire rope on hydraulic 
elevators or on cranes these numbers are exceeded. For con- 
venience of analysis, we may assume the sheaves of each block 
to be placed on separate pins as shown in Fig. 71. Beginning 
with the end of the rope fastened to the upper block, let the suc- 
cessive tensions in the parts of the rope supporting the load Q, 
be denoted by 7\, T 2 , etc.; then 

T 2 = CTx; T* = C 2 Ti 

T 4 = C*T X ; T 5 = C 4 7\; T 6 = C 5 7\ 

P = C'Ti (209) 

Q = T 6 + T 5 + T 4 + n + T 2 + 7\ 



-*K] 



(210) 



Substituting the value of 7\ from (209) in (210), we obtain 

C - 1 



p = C6 ic^rt\ Q 



Without friction, the effort required to raise the load Q is 

Q 

6 



Po 



(211) 



(212) 



Hence the efficiency for the tackle shown in Fig. 71 is 

C 6 - 1 
v 6C 6 (C-1) 



(213) 



Art. 139] 



DATA ON HOISTING TACKLE 



179 



In general when the block and tackle has n sheaves and n 
lines supporting the load Q, we get as the general expression for 
the effort 



*-*&=n\* 



and for the efficiency 



C n 



iC"(C - 1) 



(214) 



(215) 



139. Experimental Data on Hoisting Tackle.— Experimental 
data on hoisting tackle reefed with manila rope are meager, so 
in order to obtain some information as to the efficiency of such 



Table 42 — 


riOISTING 


Tackle 


R.EEFED WITH MANILA ROPE 








Block and tackle data 






Value of C 


Size of 










Ratio 
















rope 


Sheave 
diam. 


Pin 
diam. 


No. of 
sheaves 


No. of 
lines 


Q/P 


Test 


Mean 








1 


2 


1.92 


1.087 










2 


3 


2.68 


1.125 




m 


7% 


Vs 


3 

4 


4 
5 


3.37 
3.95 


1.127 
1.135 


1.13 








5 


6 


4.48 


1.13 








6 


7 


4.92 


1.14 










1 


2 


1.91 


1.098 










2 


3 


2.67 


1.125 










3 


4 


3.36 


1 . 134 




m 


m 


1 


4 
5 


5 
6 


3.93 
4.45 


1.14 
1.141 


1.14 








6 


7 


4.89 


1.143 










7 


8 


5.28 


1.143 










8 


9 


5.61 


1.143 










2 


3 


2.64 


1.136 










3 


4 


3.30 


1.142 










4 


5 


3.84 


1.155 




m 


mi 


iy s 


5 


6 


4.33 


1.155 


1.15 








6 


7 


4.72 


1.158. 










7 


8 


5.08 


1.162 










8 


9 


5.37 


1.16 










4 


5 


3.87 


1.15 










5 


6 


4.37 


1.15 




2 


13 


m 


6 


7 


4.78 


1.153 


1.15 








7 


8 ' 


5.14 


1.152 










8 


9 


5.45 


1.153 j 



180 MULTIPLE SYSTEM [Chap. VIII 

apparatus, the American Bridge Co. made an extended series 
of tests at the Pencoyd plant. These tests were made with 
standard types of manila and wire rope blocks, and an attempt 
was made to reproduce as nearly as possible actual conditions 
under which such apparatus is used in practice. The results of 
these tests were reported by S. P. Mitchell in a paper entitled 
"Tests on the Efficiency of Hoisting Tackle" and were presented 
before the American Society of Civil Engineers in September, 
1903. That part of the data pertaining to manila ropes is 
given in Table 42. In the last two columns of this table are 
given the values of C as determined by means of equation (214). 

FIBROUS TRANSMISSION ROPE 

Leather belting, while excellent for transmitting power for 
short distances under cover, is not suitable for transmitting 
power to long distances out of doors, and for this class of service, 
manila and cotton ropes are used. Cotton rope, however, is not 
used to any extent in this country. The construction of the 
manila rope used for the transmission of power is similar to that 
discussed in Art. 135. 

The transmission of power by means of manila rope gives 
satisfactory results for distances between shafts as great as one 
hundred and seventy-five feet without the use of carrying pulleys, 
while with the carriers, the distance may be increased almost in- 
definitely. Manila rope is also well-adapted to short distances. 
By the use of properly located guide pulleys power may be trans- 
mitted from one shaft to another, no matter what the relative 
positions of the shaft. There are two systems of rope driving in 
use, and each has its advocates. The two systems are com- 
monly called the Multiple or English System and the Continuous 
or American System. 

140. Multiple System. — The multiple system, which is the 
simpler of the two, uses separate ropes each spliced into an 
endless belt and running in a separate groove on each sheave 
wheel; thus each rope is absolutely independent of any other and 
carries its proportion of the load. The last statement is only 
true if the ropes are spliced carefully and the initial tension in 
each rope is made the same. The multiple system may be used 
for heavy loads and is recommended where the drive is protected 
from the weather and when the shafts are parallel or approxi- 



Art. 141] CONTINUOUS SYSTEM 181 

mately so, as in installations where the power from a prime mover 
has to be distributed to the several floors of a building. This 
system also finds favor for rolling mill service, in which service 
it is common practice to install several more ropes than are ab- 
solutely necessary to transmit the power so that the mill need 
not be closed down even if several of the ropes should fail or 
jump off. 

The advantages possessed by the multiple system are as 
follows : 

1. It is practically secure against breakdowns, and if a rope 
should break it may be removed and replaced at some con- 
venient time. 

2. The power transmitted may be increased by adding extra 
ropes. 

3. Power may be more easily transmitted to the different 
floors of an establishment. 

4. The life of a rope is greater than in the continuous system, 
since it always bends in the same direction. 

5. It is cheaper to install. 

Among the disadvantages are the following: 

1. It has more slippage than the continuous system. 

2. It is not well-adapted to quarter turn drives nor where the 
shafts are at an angle with each other. 

141. Continuous System. — In the continuous system one 
continuous rope passes around the driving and driven sheaves 
several times, in addition to making one loop about a tension 
pulley located on a traveling carriage. Since a single rope is 
used, it is evident that some device is required that will lead the 
rope from the outside groove of the driving sheave to the oppo- 
site outside groove of the driven sheave. This device is the ten- 
sion pulley. Other functions of this traveling tension pulley are 
to maintain continually a definite uniform tension in the rope, and 
to take care of the slack due to the stretching of the rope. In Fig. 
72, is shown one way of taking care of the slack by means of a 
tension carriage. 

The continuous system is well-adapted to vertical and quarter 
turn drives, and to installations having shafts that are at an 
angle to each other. It also gives better service in places where 
the rope is exposed to the weather. The following are some of 
the disadvantages: 



182 



MANILA TRANSMISSION ROPE 



[Chap. VIII 



1. A break in the rope shuts down the whole plant until the 
rope is spliced and again placed on the sheaves. 

2. All of the ropes are not subjected to the same tension; that 
is, the rope leading from the tension carriage has a greater ten- 
sion than the center ropes. 

142. Manila Transmission Rope. — For the transmission of 
power, the four or six-strand ropes are used on all sizes above 
%-inch. For the %-inch size, which is the smallest transmission 
rope made, the three-strand type gives good service. The four- 
and six-strand ropes of both hoisting and transmission types 
have the strands laid around a core which has been treated with 
a lubricant. A lubricant is used also on the inner yarns of each 




Fig. 72. 



strand, thus insuring proper lubrication of the rope. For trans- 
mission purposes experience shows that the best results are ob- 
tained when the speed of the rope is approximately 4,500 feet 
per minute. Higher speeds are used, but the life of the rope is 
decreased due to excessive wear. 

143. Sheaves. — The diameter of a sheave, used in the trans- 
mission of power by means of manila ropes, should be made 
forty times the diameter of the rope when space and speeds 
permit. Sometimes it is necessary, due to constructive reasons, 
to make the diameter less than that called for by the above rule. 
This reduction of the diameter decreases the life of the rope very 
materially and it is well to keep the minimum diameter above 
thirty-six times the diameter of the rope. 

Form of groove. — The forms of the grooves used in the two 
systems of transmission discussed in the preceding articles differ 
somewhat, although in the angle used by some of the manu- 



Art. 143] 



SHEA VES 



183 



facturers, they are similar. Experience seems to show that an 
angle of 45 degrees gives the best results for both systems. 
However, there are one or two manufacturers of rope transmis- 
sions that recommend an angle of 60 degrees. In Fig. 73 are 
shown the forms of grooves recommended for the continuous 
system, (a) and (o) being used for the driver as well as the driven, 
and (c) for the idler sheaves. As illustrated in the figure, the 
grooves are not made deep since the rope is kept taut in order to 




Fig. 73. 



decrease the tendency for it to jump out. The type of groove 
shown in Fig. 73(a) is used by the Allis-Chalmers Co.; 'for 
proportions thereof consult Table 43. For proportions of the 
form of groove used by the Dodge Mfg. Co. illustrated in Fig. 
73(6) consult Table 43. 

The form of groove commonly used in the multiple system, and 
occasionally in the continuous system, is shown in Fig. 73(d), 
and in Table 44 are given the proportions of this groove for the 
various sizes of transmission ropes. The form of the groove 
used on idlers with the multiple system is deeper than that shown 
in Fig. 73(c), but in other details it is about the same. 



184 



DIMENSIONS OF GROOVES 



[Chap. VIII 



Table 


43.- 


—Dimensions of Grooves 


FOR 


Manila Rope Sheaves 




All dimensions in inch 


es 




Allis-Chalmers standard 




Dodge Mfg. Co. standard 










rope 


Pitch 


i 


2 


3 


4 


5 


6 


Pitch 


1 


2 


3 


4 


% 
















IK 


1 


SKe 


H 


] 


H 
















IK 


IK 


K 


Ke 




l 


IK 


We 


We 


1 


K 


K 


Ke 


IK 


IK 


•He 


K 




IK 
















IK 


IK 


K 


We 


K 


m 


w 


W 


W 


We 


We 


K 


K 


IK 


IK 


We 


K 




m 
















IK 


IK 


K 


We 




m 


2 


2Ke 


IK 


We 


IK 


H 


K 


2 


IK 


1 


We 




w 


2K 


2We 


1W 6 


We 


IK 


K 


K 


2K 


2 


IK 


We 


}k 6 


2 


2% 


2% 


IWe 


1% 


IK 


1 


K 


2K 


2K 


IK 


We 



Table 44. — Dimensions of Grooves for Manila Rope Sheaves 
All dimensions in inches 









Engineers standard 






















rope 


















Pitch 


l 


2 


3 


4 


5 


6 


H 


m 


2% 


% 


% 


% 


% 


K 


Vs 


iy 2 


2^6 


IKe 


% 


V 


Vie 


K 


l 


i% 


3% 


1M 


1 


% 


Yt 


% 


lVs 


2 


3% 


1% 


1H 


r M6 


%6 


Vs 


IK 


2H 


4 


IK 


IK 


1 


% 


Vs 


l% 


2V 8 


3M 


1%6 


i% 


IK 


% 


Vs 


iy 2 


2M 


3^6 


1% 


IK 


IK 


H 


Vs 


m 


2V 2 


3% 


1^6 


i% 


IKe 


Vs 


Vs 


2 


2% 


3%6 


2K 


2 


1% 


l 


Vs 



144. Relation between Tight and Loose Tensions. — In order 
to calculate the horse power transmitted by a manila rope at a 
given speed, it is necessary to know the net tension on the ropes, 
and to get this we must determine the relation existing between 
the tight and loose tensions. Due to the wedging action of the 
rope in the groove of the sheave, the friction between the sheave 
and the rope is considerably greater than for the case of plain 
belting. The ratio between the tensions may be derived by the 
same method as that given in Art. 134. Using the same notation 
as in the discussion of the V belting, and considering a short 
length of the rope having an arc of contact subtending the angle 



Art. 144] RATIO OF ROPE TENSIONS] 185 

Ad at the center of the sheave, we get for the summation of the 
horizontal and vertical components, respectively 

AT cos -j - 2 fxN = (216) 

\a 

(2 T + AT) sin ^ - 2 iV sin - C = (217) 

Proceeding as in Art. 134, we finally obtain 
Tl -^ j«_ 

_9_ = gS in0 = ev!t (21g) 

WW 2 
1 1 — 



With the usual conditions under which manila ropes run, the 
coefficient of friction /j, may be assumed as 0.12, and the angle 
2 /3 as given in Art. 143 may be either 45 or 60 degrees. Using 

these coefficients, the values of - — ~ are as follows: 

sin j8 

For 45-degree groove, // = 0.314. 
For 60-degree groove, // = 0.24. 

.Horse power. — As in the case of belt transmission, the horse 
power is given by the formula 

H = ~(7\ - Tj (219) 

From (218), the net driving tension is given by the following 
expression : 

Tl _ Tl . [ Tl _ H?] [^i] (220 ) 

Therefore 

It is important to note that there is a rope speed that makes 
the horse power transmitted a maximum, and beyond which 
the horse power decreases. An expression for the speed corre- 
sponding to the maximum horse power may be determined by 
equating the first derivative of H with respect to v to zero, and 
solving for v. Thus from (221) 

™ . K \ Tl - »™n 

dv L a J 



186 



ANALYSIS OF A TRANSMISSION J[Chap. VIII 



whence for maximum H 



-4 



3w 



(222) 



The general form of the curve expressing the relation between 
the horse power and the rope speed is shown in Fig. 74. The 
full line applies to a 1^-inch rope running on a sheave having a 
45-degree groove, while the broken line applies to the same size 
of rope using a 60-degree groove. In plotting these graphs, it 



'25 
20 

tB 

2 
o 

0. 

15 

<u 

i_ 
o 

r ,0 
5 




































































































^— ' - 


-^ 
























/ 




"N 


s 




















/ 


r 






\ k 




















/ 








\ 




















/ 




__ — 







\, 
















/ 




s' 




" 


v 


\ 
















/ 


/ 








"v^ 


\ 
















/ 


/ 








N 


< V 














/ 














\ \ 














/ 


/ 














\ 














/ 












\ 


■\ 












/ / 
















\\ 












1 / 
















\ \ 












/ 
















\ 






































J 


















\ 










( ' 


























/ 






















M 






/■ 




























t 






















\\ 






// 




























/ 






















\\ 




/ 






















































\ 


/ 


























\ 


/ 


























\ 


/ 


























\ 



20 40 60 60 100 120 140 

Speed of Rope - ft. per sec 
Fig. 74. 



was assumed that the coefficient of friction was the same for both 
cases. 

145. Force Analysis of a Manila Rope Transmission. — As 

stated in Art. 141, one of the functions of the tension carriage is 
to produce a uniform tension in the ropes, but the following 
analysis will disclose that such a condition is not realized in the 
continuous system. In Fig. 72 is shown diagrammatically what 
is known as the American Open Drive. It should be noticed that 
the tension carriage is located just off the driving sheave. From 



Art. 145] 



ANALYSIS OF A TRANSMISSION 



187 



the discussion in Art. 137 and 144, we readily arrive at the 
following relations : 



P = 7\+ T s = (*+ l)T { 

r» 

T 3 = kT> 



(223) 



e ^ ... To = kT s e»' e > 



wr e " 

— - = pv 1 
r 7 



r 4 = kT 






p 2 n'9-2 

T 5 = kT 8 ~ 

e 2».8i 



T 7 : = kT, 



(224) 



The total net tension on the driving sheave is the difference of 
the sum of the tensions on the tight and slack sides, or 



To + 7\ + T* 



n 



(225) 



Now combining (224) with (225), the net tension T may be 
obtained in terms of T s and known constants; hence, the magni- 
tude of T 8 is fully determined since the horse power transmitted 
and the rope speed are known. Knowing T S) (223) enables us 
to establish the magnitude of the tension P. 

By comparing the expressions for To,, Ti and T G it is evident 
that these tensions are not of the same magnitude, but that each 
successive tension on the tight side is smaller than the one pre- 
ceding it. The same is true on the slack side. To overcome this 
inequality in the tension of the various ropes running over 
sheaves of unequal diameter, the above analysis shows that 
either one of the following methods could be used : 

1. By using sheaves of different materials, thus changing the 
coefficient of friction n so that ah^i = M202- 

2. By using the same material for both sheaves, but changing 
the angle of the grooves so that md\ = 1x0.62. 

The latter method is the more practical and installations using 
this scheme are in successful operation. Mr. Spencer Miller 
was probably the first one to advocate using different groove 
angles on driving and driven sheaves of unequal diameters. The 
subject was discussed by Mr. Miller in a paper read before the 



188 SHEAVE PRESSURES [Chap. VIII 

American Society of Civil Engineers in June, 1898, and reported 
in volume 39, page 165 of the Transactions of that society. 

146. Sheave Pressures. — The series of equations given by 
(224) above enables us to determine the approximate pressures 
coming upon the shafts of the sheaves due to the rope tensions. 
The pressure upon the shaft of the driving sheave, assuming the 
tight and slack side to be practically parallel, is 

Qi = T, + T 3 + T, + T 6 + T 6 4- T 7 (226) 

The pressure upon the shaft of the driven sheave is 

Q 2 = T 1 + T % + T 3 + T, + T h + T 6 (227) 

The pressures upon the shafts of the idler sheaves a and b are 
respectively 

Q 3 = T 7 + T s , (228) 

and 

Q 4 = 7\ + T 8 (229) 

The horse power absorbed by the friction of the bearings on the 
shafts, due to the pressure just determined, is considerable and 
may be estimated by the following expression : 

H, = Q^iQiNm + Q 2 N 2 r 2 + Q 3 N 3 r 3 + Q 4 iV 4 r 4 ), (230) 

in which JV denotes the number of revolutions per minute of 
the sheave, r the radius of the sheave shaft, and /jl 3 the coefficient 
of journal friction. 

147. Sag of Rope. — In practically all rope transmissions it is 
important to determine the approximate sag of the ropes. In 
arriving at a formula by means of which the probable sag may 
be estimated, no serious error is introduced by assuming that the 
rope hangs in the form of a parabola instead of a catenary. In 
Fig. 75 is shown a rope suspended over two sheaves, the line 
ABC representing approximately the curve assumed by the rope. 
From the equation of the parabola we have 

I = I ^ 

Substituting the value of L 2 = L — L x in (231) and reducing 
the expression to the simplest form, we finally get 

Vhx + Vh~2 



Art. 147] 



SAG OF ROPE 



189 



In a similar manner 



U 



L-y/hn 



The horizontal tension in the rope at the lowest point B is 



(233) 



wLi wh\ 



2Ai 



2 hi 



(234) 



The difference in the tensions at any two points of a rope form- 
ing a catenary is equal to the difference in elevation of these 
points multiplied by the weight per unit length of rope. Treat- 
ing the rope ABC in Fig. 75 as if it formed a catenary and applying 
the property just mentioned, the tension T a at A is 



T a = T + whi 



[ft**] 



(235) 




Fig. 75. 



and the tension at C is 



T c = T + W h 2 = W [^ 2 + h 2 ] 



(236) 



From (235), it follows that the magnitude of the sag hi is given 
by the following expression : 



fci 



^{T a ± VtI -2L\ 



w 2 ) 



(237) 



and from (236), the sag h 2 is 
1 



h2 = t»( T < +^T 2 c -2 LW) (238) 

By means (237) and (238) the sag of the ropes on either the 
tight or slack side of the transmission may be estimated by sub- 



190 EFFICIENCY OF ROPE DRIVES [Chap. VIII 

stituting the proper values for the tension. From an inspection 
of (237), it is evident that for the same tension T a in the rope at 
A there are two different values of h x ; however, in rope-trans- 
mission problems the smaller value is the correct one to use. 
The statement applies equally well to (238). 

It is important to note that the above discussion applies to the 
rope standing still. The sag of a rope, transmitting power may- 
be determined approximately by means of (234) by substituting 
the proper value of the tension T. 

A special formula may be deduced for the case in which the 
transmission is horizontal having sheaves of the same diameter. 

By substituting for L x = ~ in either (237) or (238), the amount 

of sag h is given by the following expression : 



-ifc±Jr-¥\ 



A = 24 r± v !_ ^J (239) 

In general the bottom rope should form the driving side, as 
with this arrangement the sag of the slack rope on top increases 
the arc of contact. 

148. Efficiency of Manila Rope Drives. — The efficiency of 
manila rope transmission is generally high according to several 
series of experiments performed both in this country and abroad. 
During the latter part of 1912, the Dodge Mfg. Co. of Misha- 
waka, Ind. conducted a series of experiments to obtain some 
information relating to the efficiencies of four general plans of 
manila rope driving. The four plans investigated were as 
follows : 

1. Open drive using the American or continuous system, as 
shown in Fig. 72. 

2. Open drive using the English or multiple system. 

3. American "up and over" drive. 

4. English "up and over" drive. 

In the tests upon these various plans of rope driving, from 
one- to eight-rope, operating at speeds ranging from 2,500 to 
5,500 feet per minute were used. High-grade manila ropes one 
inch in diameter, treated with a rope dressing so as to make them 
moisture-proof and to preserve the surface, were used throughout 
the tests. The sheave grooves were in accordance with accepted 
Dodge practice, namely a 60-degree angle for the American sys- 
tem and a 45-degree angle for the English system. All idler 



Art. 148] 



EFFICIENCY OF ROPE DRIVES 



191 



sheaves used in the various arrangements were provided with 
U-shaped grooves. 




2500 



3000 



3500 4000 4500 

Rope Speed in ft. per min. 
Fig. 76. 



5000 



Altogether about seven hundred tests were made, the general 
results of which were published in a paper presented by Mr. E. 
H. Ahara before the American Society of Mechanical Engineers. 



+- 

c 

V 

u 

s_ 

CD 
D. 

C 

c 

O 
? 

UJ 


100 
90 
80 
70 
60 
RO 
















































, 




< 


Amer. Open 




' 


















































Eng.Open 








^- — 


























AmerU.&O 




















y 


















/ 














Eng U.&0 




/ 


y 














V 




















sA 



































































































































































































Number of Ropes 
Fig. 77. 



An analysis of the results published seemed to indicate that the 
efficiency for low rope speeds was higher than that obtained at 
the high rope speeds. This result is shown clearly in Fig. 76, 



192 SELECTION OF ROPE [Chap. VIII 

which represents the results obtained from the tests on both 
systems of open drive operating with six ropes at three-quarters 
load, the distance between the centers of the sheaves being fifty 
feet. Furthermore, the tests showed that the efficiency was not 
affected materially by varying the distances between the driving 
and driven sheaves. The tests also showed that the efficiency at 
half load was but very little less than that obtained at full load. 
For the size of rope used in the experiments, namely one inch, 
the American system had considerable more capacity as well as 
a higher efficiency than the English system. In Fig. 77 is repre- 
sented the relation existing between the efficiency and the num- 
ber of ropes used for the four plans of driving. 

149. Selection of Rope. — Manila ropes for transmission pur- 
poses are seldom less than one inch in diameter, and due to the 
resistance offered to bending over the sheaves, ropes exceeding 
one and three-quarter inches in diameter are not in general use. 
For heavy loads such as are met with in rolling-mill installations, 
ropes two inches in diameter and larger are used. 

In order to arrive at the proper number and size of ropes 
required to transmit a given horse power, the size of both the 
driving and driven sheaves should be decided, as the smallest 
sheave in the proposed installation will determine in a general 
way the largest rope that may be used. If possible, the diame- 
ters of these sheaves should be such that the rope will operate at 
somewhere near its economical speed, which, as stated in Art. 
142, has been found in practice to be about 4,500 feet per minute. 
To obtain a reasonable length of service from a given rope, its 
diameter should not exceed one-fortieth of the diameter of the 
smallest sheave. According to the American Manufacturing Co. 
of Brooklyn, N. Y., it is considered good practice to use a small 
number of large ropes instead of a large number of small ropes, 
notwithstanding the fact that the first cost of the sheaves for the 
former exceeds that required for the smaller ropes. In an instal- 
lation using a small number of large ropes the number of splices 
is smaller; hence, the number of shutdowns due to the failure 
of splices is decreased; furthermore, since the large rope has a 
greater wearing surface, its life is increased. 

150. Cotton Rope Transmission. — The transmission of power 
by means of cotton rope is not used to any extent in this country, 
but in England it is used extensively in all kinds of installations. 



Art. 150] COTTON ROPE 193 

The strength of good cotton rope is about four-sevenths of that of 
high-grade manila rope, and its first cost is about 50 per cent, 
more. Due to the soft fiber, the cotton rope is more flexible than 
the manila rope, and for that reason smaller sheaves may be used 
for the former. According to well-established English practice, 
the diameters of the sheaves are made equal to thirty times the 
diameter of the rope. The cotton rope, as generally used, is com- 
posed of three strands, and being somewhat soft, it is wedged into 
the grooves of the sheave. 

According to some of the American rope manufacturers, a 
manila rope of a given size will transmit considerably more power 
than the same size of cotton rope. In view of this statement 
it is interesting to compare the power that a given size of both 
manila and cotton rope, say 1% inches in diameter, will trans- 
mit at a speed of 4,500 feet per minute. According to a well- ' 
known American manufacturer, the manila rope under the 
above conditions will transmit 29.1 horse power. According to 
a table published by Edward Kenyon in the Transactions of the 
South Wales Institute of Engineers, a l^-inch cotton rope will 
transmit 33.4 horse power at the same speed. This result 
represents an increase of 14.7 per cent, in the power transmitted, 
and also indicates that higher tensions are permissible with 
cotton rope. As stated above, cotton rope is not as strong as 
manila rope; hence, these higher tensions must be due to the 
structure of the rope. The fibers of cotton rope being soft and 
more flexible do not cut or injure each other when the rope is 
subjected to bending under a tension, as is the case with the 
manila fiber; the grooves of the cotton rope sheave are so formed 
that the rope is wedged into the groove angle; hence, the effect of 
centrifugal force is not so marked as with manila rope transmis- 
sion. The inference is clear that it is possible to employ high 
speeds with cotton rope ; and such is the case, as English manu- 
facturers recommend speeds up to 7,000 feet per minute. 

References 

The Constructor, by F. Retjleaux. 

Rope Driving, by J. J. Flather. 

Machine Design, Construction and Drawing, by H. J. Spooner. 

Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 

Rope Driving, Trans. A. S. M. E., vol. 12, p. 230. 

Working Loads for Manila Ropes, Trans. A. S. M. E., vol. 23, p. 125. 



194 REFERENCES [Chap. VIII 

Efficiency of Rope Drives, Proc. The Eng'g Soc. of W. Pa., vol. 27, No. 3, 
p. 73. 

Efficiency of Rope Driving, Trans. A. S. M. E., vol. 35, p. 567. 

Transmission of Power by Manila Ropes, Power, May 12, 1914 (vol. 39, 
p. 666). 

Transmitting Power by Rope Drives, Power, Dec. 8, 1914, (vol. 40, p. 808) 

The Blue Book of Rope Transmission, American Mfg. Co. 



CHAPTER IX 
WIRE ROPE TRANSMISSION 

The present-day application of wire rope is chiefly to hoisting, 
haulage, and transporting service, and but little to the actual 
transmission of power. In this chapter, wire rope will be dis- 
cussed under two general subheads as follows: (a) wire rope 
hoisting, and (&) wire rope transmission. 

WIRE ROPE HOISTING 

For haulage service, the six-strand seven-wire rope, generally 
written 6 X 7, is used, while for hoisting a 6 X 19, 8 X 19, or 
6 X 37 construction is employed. The rope last mentioned is 
the most flexible and may be used with smaller sheaves than 
either of the others, but the wires are much smaller; hence it 
should not be subjected to excessive external wear. The 6 X 19 
and 8 X 19 ropes are recommended for use on cranes, elevators 
of all kinds, coal and ore hoists, derricks, conveyors, dredges, and 
steam shovels. The 6 X 37 rope, which is extra flexible, is used 
on cranes, special hoists for ammunition, counterweights on 
various machines, and on dredges. 

A hoisting rope under load is subjected to the following prin- 
cipal stresses: 

(a) Stresses due to the load raised. 

(6) Stresses due to sudden starting and stopping. 

(c) Stresses due to the bending of the rope about the sheave. 

(d) Stresses due to slack. 

151. Relation between Effort and Load. — In hoisting machinery 
calculations, it is necessary to know the relation existing between 
the effort and the resistance applied to the ends of the rope run- 
ning over a sheave. The rigidity of the rope and the friction of 
the sheave pin increase the resistance that the effort applied to 
the running off side must overcome. By applying the same line 

195 



196 BENDING STRESSES [Chap. IX 

of reasoning as used in Art. 137, we obtain a relation which is 
similar to (207), namely 

The efficiency of the ordinary guide sheave, obtained by apply- 
ing the usual definition of efficiency, is as follows : 

V = jj (241) 

152. Stresses Due to Starting and Stopping. — A rope whose 
speed changes frequently, as in the starting and stopping of" a 
load, is subjected to a stress which in many cases should not be 
neglected. This stress depends upon the acceleration given to 
the rope, and its magnitude is determined by the well-known 
relation, force is equal to the mass raised multiplied by the 
acceleration. In the calculation of the size of rope for mine 
hoisting or for elevator service, the stress due to acceleration 
assumes special importance. 

153. Stresses Due to Bending. — The stresses due to the bend- 
ing of the rope about sheaves and drums are of considerable 
magnitude and should always be considered in arriving at the 
size of a rope for a given installation. Several formulas for calcu- 
lating these stresses have been proposed by various investigators, 
but they are all more or less complicated. The simplest of these 
is the following : 

S b = E±, (242) 

in which D represents the pitch diameter of the sheave, E the 
modulus of elasticity of the rope, Sb the bending stress per square 
inch of area of wires in the rope, and 8 the diameter of the wire in 
the rope. This formula was adopted by the American Steel and 
Wire Co. To determine the value of E the company conducted 
a series of experiments on some six-strand wire rope having a 
hemp center. This investigation seemed to show conclusively 
that the modulus of elasticity for a new rope does not exceed 
12,000,000. Using this value in (242), a series of tables was 
calculated and published in the company's Wire Rope Hand 
Book. From these data the curves shown in Figs. 78, 79 and 
80 were plotted. They show the relation between the bending 



Art. 153] 



BENDING STRESSES 



197 





















\ 














\ 
























































, 












\ 


























\ 














\ 


BENDING STRESSES 
























\ 














\ 




14000 




















\ 














\ 
























\ 














\ 


IN 








































V 


























\ 














\ 


. 6x7 ROPES 










































\ 


tor 




























\ 














\ 




























\ 


















VARIOUS SHEAVE DIAMETERS 


















\ 










\ 
















\ 


















\ 












v_ 














\ 




















V 




























\ 




















\ 












\ 
















\ 


































, 
















\ 


































\ 
















S 






















V 






























\ 






















\ 














\ 
























11000 


















\, 












s 


















\ 




















V 














\ 










































































\ 
























L 
































s 








































s: 


























10000 




















\ 


































\ 












\ 












\ 


































\ 
















\ 












\ 














A 






















. 




























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s 




















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V 




































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9000 
























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BENDING STRESSES 



[Chap. IX 



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Art. 153] 



BENDING STRESSES 



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Sheave Diameters in Inches 
Fig. 80. 



200 



STRESSES DUE TO SLACK 



[Chap. IX 



stresses and various diameters of sheaves or drums for the more 
common sizes of 6 X 7, 6 X 9, and 6 X 37 wire ropes. How- 
ever, instead of using the ton as a unit, all stresses are reduced 
to pounds. 

154. Stresses Due to Slack. — In any kind of hoisting operation 
it is important that the rope shall have no slack at the beginning 
of hoisting, else the load will be suddenly applied and the stress 
in the rope will be much in excess of that due to the load raised. 
The results of various dynamometer experiments in this connection 
are exhibited in Table 45. 

Table 45. — Tensions Due to Slack as Shown by Dynamometer 



Slack in inches 


Weight of cage and load 


3,672- 


6,384 


11,312 


11,310 




5 

6 

12 


4,032 

5,600 

8,960 

12,520 


6,720 
11,200 
12,320 
15,680 


11,542 
19,040 
23,520 
28,000 


11,525 
19,025 
25,750 
28,950 



The theoretical relation between the tension in the rope and the 
load raised may be deduced as follows: 

Let W = load to be raised. 

T = tension in the rope corresponding to the maximum 

elongation. 
a = acceleration of the rope at the beginning of hoisting. 
b = elongation of the rope due to the load W. 
c = maximum elongation of the rope. 
e = amount of slack in the rope. 

The raising of the rope through the distance e, so as to take up 
the slack, may be considered as producing the same effect as 
dropping the load W through the distance e, assuming the 
acceleration in both cases as constant and equal to a. Letting 
v denote the velocity at the instant when the slack e is taken up, 
we have v 2 = 2 ae. From this it follows that the kinetic energy 
of the load W at the instant the rope is taut, is 
Wv 2 = Woe 
2 a a 



(243) 



Due to this loading the rope elongates a distance c, the final 
tension being T. Hence W in moving through this distance c 



Art. 154] STRESSES DUE TO SLACK 201 

does work equal to Wc. Immediately preceding the elongation 
of the rope, the tension therein is zero and at the end of the 
elongation the tension has a magnitude T; therefore, the work 
of the variable tension during the period of rope elongation is 

Tc 

-jr. To do this internal work, the load has given up its kinetic 

Wae 
energy and the work Wc; hence 

Tc Wae . T „ lnAA . 

-=■ = —— + Wc (244) 

z 9 

Assuming that Hooke's Law will hold approximately in the 

case of a rope, we get 

Wr 

T=-^ (245) 

Substituting this value of T in (244), and solving for c, we 
finally get 

c = b ± b ^1 + ^ (246) 

The conditions of the problem indicate that the positive sign 
is the proper one; hence, substituting the value of c in (245), 



-w[i + ^ 



, , 2ae 



(247) 



If the slack e is zero (247) shows that T = 2 W; that is, the 
tension is double the load, which fact was established in Art. 18. 
The amount of slack simply has the effect of increasing the ratio 

T 

tt?, which, as shown, cannot be theoretically less than two. 

It is not to be expected that experiments would give exactly the 
theoretical values, on account of the fact that wire rope differs 
materially from a rigid rod, and a certain amount of stretch not 
according to Hooke's Law will come into play before the actual 
elongation of the material begins. This fact in a measure, re- 
lieves the "suddenness," so to speak, of the action, and we would 
expect the tensions measured by the dynamometer to be less than 
those given by (247) . To get the experimental values by means 
of (247), it will be necessary to introduce a coefficient K in the 
equation, making it 

T = W I; +KJ1 +^ (248) 



[ 1 + ^ 1 + ^] 



This coefficient must of course be determined by experiments, 
and will doubtless vary with the construction of the rope and quite 



202 SELECTION OF ROPE [Chap. IX 

likely with the load W and the slack e. Unfortunately, in the 
experiments quoted in Table 45 no attempt was made to de- 
termine the acceleration of hoisting, and as a consequence, one 
essential factor is lacking; hence it is impossible to arrive at 
probable values of the coefficient K unless an assumption regard- 
ing the ratio a to & is made. 

155. Selection of Rope. — The maximum stress coming upon a 
rope is the summation of the separate stresses that may be 
present in any installation. These separate stresses have been 
discussed in the preceding articles, and having determined their 
intensities, the magnitude of the maximum is readily obtained. 
The next step is to determine the ultimate strength of the prob- 
able size of rope to be used, by multiplying the maximum stress 
by a factor commonly called the factor of safety. This factor 
varies with the class of service for which the rope is intended, and 
the following values may serve as a guide in the solution of wire 
rope problems: 

For elevator service the factor of safety varies from 8 to 12. 

For hoisting in mines the factor of safety varies from 2}i to 5. 

For motor driven cranes the factor of safety varies from 4 to 6. 

For hand power cranes the factor of safety varies from 3 to 5. 

For derrick service the factor of safety varies from 3 to 5. 

Having calculated the ultimate strength, select the size of rope 
that is strong enough. In practically all hoisting rope calcula- 
tions, it will be found that two or more wire ropes of different 
sizes and quality will satisfy the conditions of the problem; for 
example, from Table 46 it is evident that a %-inch crucible 
steel rope and a %-inch plow steel rope of the 6 X 19 construc- 
tion have the same ultimate strength; hence, either of these ropes 
could be selected. In the example just quoted, the ^-inch plow 
steel rope would be preferable to the %-inch crucible steel rope, 
since the smaller sheave called for by the former size would effect 
a saving of space as well as in the first cost. In a preceding 
paragraph, the uses of the various types of wire rope were dis- 
cussed briefly. In Table 46 is given information pertaining 
to the ultimate strengths and weights of rope, as well as the 
minimum diameter of sheaves recommended by the manufacturer. 

156. Hoisting Tackle. — The analysis of blocks and tackles 
reefed with wire rope is similar to that given in Art. 138 for manila 
rope, and the formulas deduced there also apply in the present 
case, provided a proper value is assigned to the coefficient C. 



Art. 156] 



WIRE ROPE TABLES 
Table 46. — Steel Wire Rope 



203 





i meter 
inches 


6X7 construction 


6 X 19 construction 


Di 

in 


Weight 
per 

foot 


Mini- 
mum 
sheave 
diam. 


Ultimate 


strength 


Weight 
per 

foot 


Mini- 
mum 

sheave 
diam. 


Ultimate strength 




Crucible 
steel 


Plow 
steel 


Crucible 
steel 


Plow 

steel 


K 












0.10 


12 


4,400 


5,300 




«6 


0.15 


27 


7,000 


8,800 


0.15 


15 


6,200 


7,600 


K 




0.22 


33 


9,200 


11,800 


0.22 


- 18 


9,600 


11,500 




Ke 


0.30 


36 


11,000 


14,000 


0.30 


21 


13,000 


16,000 


V?. 




0.39 


42 


15,400 


20,000 


0.39 


24 


16,800 


20,000 




%6 


0.50 


48 


20,000 


24,000 


0.50 


27 


20,000 


24,600 


H 




0.62 


54 


26,000 


32,000 


0.62 


30 


25,000 


31,000 




H 


0.89 


60 


37,200 


46,000 


0.89 


36 


35,000 


46,000 


H 




1.20 


72 


48,000 


62,000 


1.20 


42 


46,000 


58,000 




l 


1.58 


84 


62,000 


76,000 


1.58 


48 


60,000 


76,000 


IK 




2.00 


96 


74,000 


94,000 


2.00 


54 


76,000 


94,000 




IK 


2.45 


108 


92,000 


120,000 


2.45 


60 


94,000 


116,000 


IK 




3.00 


120 


106,000 


144,000 


3.00 


66 


112,000 


144,000 




iy 2 


3.55 


132 


126,000 


164,000 


3.55 


72 


128,000 


164,000 


m 












4.15 


78 


144,000 


188,000 




m 










4.85 


84 


170,000 


224,000 


IK 












5.55 


96 


192,000 


254,000 




2 










6.30 


96 


212,000 


280,000 


2K 












8.00 


108 


266,000 


372,000 




2K 










9.85 


120 


340,000 


458,000 


Wx 












11.95 


132 


422,000 


550,000 





imeter 
inches 


8 X 19 construction 


6 X 37 construction 


Di 
in 


Weight 
per 
foot 


Mini- 
mum 
sheave 
diam. 


Ultimate strength 


Weight 
per 
foot 


Mini- 
mum 
sheave 
diam. 


Ultimate strength 




Crucible 
steel 


Plow 
steel 


Crucible 
steel 


Plow 
steel 


K 




0.20 


12 


9,320 




0.22 


12 


9,300 


10,600 




Ke 


0.27 


14 


12,600 




0.30 


14 


12,700 


15,000 


K 




0.35 


16 


16,000 


19,000 


0.39 


16 


16,500 


19,500 




He 


0.45 


18 


20,200 


24,000 


0.50 


18 


21,000 


25,000 


K 




0.56 


21 


24,800 


30,000 


0.62 


21 


25,200 


32,000 




% 


0.80 


22 


35,200 


44,000 


0.89 


22 


38,000 


46,000 


H 




1.08 


26 


46,000 


56,000 


1.20 


26 


50,000 


58,000 




1 


1.42 


30 


59,400 


72,000 


1.58 


30 


64,000 


74,000 


IK 




1.80 


34 


76,000 


92,000 


2.00 


34 


78,000 


92,000 




IK 


2.20 


38 


94,000 


112,000 


2.45 


38 


100,000 


116,000 


IK 




2.70 


42 


114,000 


136,000 


3.00 


42 


122,000 


142,000 




m 


3.19 


45 


132,000 


160,000 


3.55 


45 


142,000 


168,000 


IK 


m 










4.15 
4.85 




158,000 
190,000 


190,000 
226,000 


IK 


2 










5.55 
6.30 




212,000 
234,000 


250,000 
274,000 


2K 


2K 










8.00 
9.85 




300,000 
374,000 


368,000 
450,000 


m 












11.95 




466,000 


556,000 



204 



HOISTING SHEAVES ■ 



[Chap. IX 



Experimental data on wire-rope hoisting tackle. — Some years 
ago the American Hoist and Derrick Co. of St. Paul, Minn., 
conducted a series of experiments on three standard sizes of 
blocks reefed with wire rope. The results of these tests are given 
in Table 47. By using the relation between P and Q in 
terms of C for the various combinations listed, it is possible to 
calculate the value of C. This was done by the author and the 
values are tabulated in the last two columns of Table 47. 




Fig. 81. 

An inspection of the values of C given in this table shows that for 
a given size of rope the coefficient C may safely be assumed as 
constant. 

157. Hoisting Sheaves and Drums. — (a) Sheaves. — The sheaves 
used for hoisting purposes vary considerably in their design. 
For crane work the sheaves are usually constructed with a central 
web in place of arms and in order to reduce the weight, openings 
may be put into this web. Such a sheave is shown in Fig. 81 
and in Table 48 are given some of the leading dimensions 
pertaining to the design shown in Fig. 81. As a rule, sheaves of 
this class are bushed with bronze or some form of patented bush- 
ing, and run loose on the pin. For very heavy crane service, the 
sheaves are frequently made of steel casting, cast iron being 
used for the medium and lighter class of service. 



Art. 157] EXPERIMENTAL DATA ON HOISTING TACKLE 205 
Table 47. — Hoisting Tackle Reefed with Wire Rope 





Block and tackle data 

■ 


Ratio 

P/Q 


Value of c 


rope 


Sheave 
diam. 


Pin 
diam. 


No. of 
sheaves 


No. of 
lines 


Test 


Mean 




9 


m 


1 


2 


0.518 


1.075 






2 


2 
3 


0.559 
0.358 


1.078 
1.076 




X 


3 


3 

' 4 


0.385 

0.278 


1.076 
1.076 


1.076 


4 


4 
5 


0.298 
0.230 


1.075 
1.076 




5 


5 
6 


0.247 
0.198 


1.076 
1.076 






6 


6 


0.213 


1.076 






H« 


iy 2 


1 


2 


0.516 


1.068 






2 


2 
3 


0.549 
0.355 


1.066 
1.066 




% 


3 


3 

4 


0.376 
0.273 


1.063 
1.063 


1.064 


4 


4 
5 


0.291 
0.225 


1.064 
1.063 




5 


5 

6 


0.240 
0.193 


1.064 
1.064 






6 


6 


0.206 


1.064 






13% 


ih 


1 


2 


0.513 


1 . 053 






2 


2 
3 


0.541 
0.351 


1.055 
1.054 




3 A 


3 


3 

4 


0.369 
0.270 


1.053 
1.054 


1.054 


4 


4 
5 


0.284 
0.221 


1.053 
1.054 




5 


5 
6 


0.233 
0.189 


1.054 
1.054 






6 


6 


0.199 


1.053 





For heavy high-speed hoisting as found in mining operations, 
the arms consist of steel rods cast into the hub and rim, as shown 
in Fig. 82. Sheaves of this class are not bushed as in crane service, 
but are keyed to the shaft. 

The grooves of all hoisting sheaves should be finished smooth 



206 HOISTING SHEAVES [Chap. IX 

Table 48. — General Dimensions of Wire Rope Sheaves 





Dimensions in inches 




Size of 






rope 




























1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


ll 


12 




m 


H 

H 


1 


%6 


K 

5 Ae 


% 
% 






% 

K 








IK 




1% 








% 


2 


%6 


IK 


Kb 


Kb 


% 


1% 


m 


Kb 


2% 


1 


K 


% 


2M 


% 


1% 


k 


% 


% 


IK 


2 


% 


2^ 


IK 


K 


i 


2M- 


Vie 


2 


%6 


M 


1 


m 


2K 


% 


2% 


IK 


K 


m 


2% 


Ke 


2M 


% 


Kb 


1Kb 


2-2K 


2K 


% 


3M-4 


IK 


l 



so as to protect the individual wires of the rope. The radius 
of the bottom of the groove should be made slightly larger than 
the radius of the rope, so that the latter will not be wedged into 




Fig. 82. 

the groove. It is important that the alignment of sheaves be the 
best possible, otherwise the rope will slide on the sides of the 
groove and cause an undue amount of wear on both the rope and 
sheave. The diameter of the sheave should be made as large 
as possible to keep down the bending stresses. In Table 46 
are given the minimum sheave diameters recommended by the 
wire rope manufacturers. It is customary for crane builders to 



Art. 157] 



HOISTING DRUMS 



207 



use much smaller sheaves. By using sheaves having a diameter 
of from eighteen to twenty times the diameter of the rope, a 
considerable saving in space may result but at the same time 
the life of the rope is decreased materially. 

(b) Drums. — In hoisting machinery, the drums are usually 
grooved to receive the rope and their lengths should be sufficient 
to hold the entire length of rope in a single layer. The use of the 
plain ungrooved drum should be avoided unless it is lagged. If 
the drum is grooved, the pitch of the grooves must be made 
slightly larger than the diameter of the rope so that the successive 
coils do not touch when the rope is wound onto the drum. In 
Fig. 83 is shown the form of groove used by several crane builders, 




Fig. 83. 



and in Table 49 are given the various dimensions required to 
lay out these grooves. 

Table 49. — Dimensions of Grooves for Wire Rope Drums 











Size of wire rope 


























sion 




















% 


He 


M 


He 


% 


Z A 


% 


1 


1 


He 


y* 


He 


% 


% 


13 Ae 


15 Ae 


IHe 


2 


K 2 


K 


Ha 


Vie 


% 


% 


% 


% 


3 


%2 


%4 


Vs 


%4. 


%2 


He 


VZ2 


H 



The diameters of the drums are usually made larger than those 
of the sheaves for a given size of rope in order to keep down the 
length to a reasonable dimension. In general, the diameters of 
crane drums vary from twenty to thirty times the diameter of 
the rope. The speed of hoisting, the load to be raised, and the 
life of the rope should be considered in arriving at the proper 
diameter of the drum. In order to relieve the rope anchor on 
the drum, always add about two extra coils to the calculated 



208 



CRANE DRUMS 



[Chap. IX 



number, so that the extra coils of rope remain unwound on the 
drum. 

158. Design of Crane Drums. — A simple design of a plain 
hoisting drum running loose on the shaft is shown in Fig. 84. 




Fig. 84. 

The shaft a is driven by means of the gear b, to the web of which 
are bolted the double conical friction blocks c. These blocks fit 
into the clutch rim, which in this case is integral with the drum 






3 


' 


k 










w5/////M 








I 












•. 


- 


h 

s: 




r~ 


' 


: a 










. ife 


) 


B 1 








\| 




^ 


^ 


1 




— e 


- 








^ 


^ 






6 




w L 


D 


N 





Fig. 85. 

d. To rotate the drum with the gear, the clutch is engaged by- 
sliding the drum along the shaft a by means of the operating 
mechanism shown at the left. This design is used on light hoist- 
ing engines manufactured by the Clyde Iron Works of Duluth, 
Minn, 



Art. 159] CONICAL DRUMS 209 

A good design of a crane drum is shown in Fig. 85. In this 
case the shaft h is held stationary, the drum hubs being bushed 
with bronze as shown. The driving gear m is keyed rigidly to 
the drum h, which in this case is scored for a hoisting chain al- 
though the same design of drum may be used with rope. Fre- 
quently the shaft, instead of being stationary, is cast into the 
drum and the whole combination rotates on the outer bearings. 

The correct stress analysis for a hoisting drum is a complicated 
problem, and the following approximate method is generally used 
in arriving at, or for checking, the thickness of the metal below 
the bottom of the groove: 

1. Determine the bending stresses by treating the drum as a 
hollow cylindrical beam supported at the ends. Assume the 
maximum rope loads as concentrated at or near the middle, 
depending upon the scoring on the drum. 

2. Determine the crushing stress due to the tension in the coils 
of rope about the drum. The rope tension varies from coil to 
coil, and since maximum values are sought, consider only the 
first coil, namely, the one supporting the load. 

3. Determine the shearing stress due to the torsional moment 
transmitted. As a rule this stress is very small and is usually not 
considered. 

4. Combine the stresses calculated in (1) and (2) above. 
Drums thus designed have sufficient strength, and in general the 
weight is not excessive. 

159. Conical Drums. — In mine hoists, it is a usual practice to 
employ drums having varying radii for the successive coils of the 
rope. The object of such an arrangement is to obviate the varia- 
tions in the load on the drum due to the varying length of rope. 
Theoretically, the net moment of the rope pull about the drum 
axis should be a constant in order that the motors or engines 
coupled to the drum may operate economically. This condition 
would require a drum of curved cross-section, a form that would 
be difficult to construct. In practice, the section of each half of 
the drum is given the form of a trapezoid, and for that reason it 
is possible to balance the moments on the drum at but two points 
of the hoist, namely at the top and bottom. 

(a) Relation between R 2 and Ri. — To determine the relation ex- 
isting between the large and small diameters R 2 and R x of the 
drum, so as to fulfil the condition just mentioned, we may pro- 
ceed as follows : 



210 CONICAL DRUMS [Chap. IX 

Let C = weight of cage and empty car. 
H = depth of mine in feet. 
Q = weight of ore in car. 
w = weight of the hoisting rope, pounds per foot. 

Neglecting the inertia forces, the moment of the rope tension 
at the beginning of the hoisting period is 

M x = (C + Q + wH)(l + /i)22i - C(l - riRi, (249) 

in which the symbol n represents a friction coefficient that may 
be assumed as equivalent to 0.05 for vertical mine shafts. 
The moment at the end of a trip is 

M 2 = (C + Q)(l + m)# 2 - (C + wH)(l - /Offx (250) 

Equating these moments and solving for the radius of the drum 
at the large end, we find 

*- pisses* 9 ]*--* (251) 

Evidently the greater the depth of the mine shaft, the greater 
is wH relative to Q and C, and the greater the value of the 
factor m. 

(&) Length of the conical drum. — The conical drum must be 
provided with spiral grooves to receive the rope, and the number 
required to hoist from a depth H is 

H 



7T(W + l)Ri 



(252) 



Several extra turns are required, so that at the beginning of hoist- 
ing the rope will be coiled several times around the drum. The 
same number should be added at the end of hoisting. This num- 
ber of extra turns is fixed by state mining laws. 

If L denotes the length of the drums and p the horizontal pitch 
of the grooves, then 

L = \ , 5n P + n '] V, (253) 

Ltt(w + i)Rx A ' 

in which n' represents the extra number of coils added. 

The length of a conical drum is necessarily great, and for that 
reason the drum must be located at a considerable distance from 
the mine shaft to reduce as much as possible the angular dis- 
placement of the rope from the center line of the head sheave. 



Art. 160] FLAT WIRE ROPES 211 

This displacement is called the fleet angle and should not exceed 
one and one-half degrees on each side of the center line, or a total 
displacement of three degrees. When it is impossible to locate 
the drum far enough back from the head sheave to keep the 
fleet angle within these limits, it is necessary to guide the rope 
onto the head sheaves by means of rollers or auxiliary sheaves, 
(c) Composite drum. — For deep mines, another form of drum 
called the composite drum is frequently substituted for the plain 
conical type. This consists of a cylindrical center portion and 
conical ends. One rope is wound from one end up the cone and 
over the cylindrical portion, while the other is unwound from the 
cylindrical part and down the other cone. This form of drum 
has the advantage of decreased diameter and shorter length, but 
possesses the disadvantage of not entirely balancing the effect of 
the rope. 

160. Flat Wire Ropes. — In the preceding articles, the round 
wire rope has been discussed more or less in detail, and the various 
points brought out are applicable in general to the flat rope. 
This type of rope consists of a number of round wire ropes, called 
flat rope strands, placed side by side. Its principal uses are for 
mine hoisting; for operating emergency gates on canals; for oper- 
ating the spouts on coal arid ore docks; and in elevator service 
for counterbalancing the hoisting ropes. The individual strands, 
composed of four separate strands containing seven wires each, 
are of alternate right and left lay and are sewed together with 
soft Swedish iron or steel wire. The sewing wires, being much 
softer than the wires that compose the strands, serve as a cushion 
for the strand and at the same time will wear out much faster. 
Flat wire rope with worn out sewing may be resewed with new 
• wire, and in case any particular strands are damaged, they may be 
replaced by new ones. Flat ropes are made in thicknesses vary- 
ing from yi inch to % inch, and widths ranging from 1% to 8 
inches. The material used in the construction of flat ropes may 
be either crucible cast steel or plow steel, the former being more 
common. The following are some of the advantages flat ropes 
possess over round ropes. 

1. In hoisting from deep mines it is desirable to use a rope that 
has no tendency to twist and untwist. This tendency is obvi- 
ated by the use of a flat rope. 

2. The reels required for coiling up the flat rope occupy less 
space and are much lighter and cheaper to construct than large 



212 SINGLE LOOP SYSTEM [Chap. IX 

cylindrical and conical drums. The decrease in bulk and weight 
is especially important when the mines are located in places 
accessible only by pack train. 

At the present time flat ropes are used but little for mine hoist- 
ing and hence the field of application of such ropes is more or less 
restricted. 

WIRE ROPE TRANSMISSION 

Wire rope as a medium for transmitting power is used where 
the distances are too great for manila ropes. The recent develop- 
ment of electrical transmission is gradually crowding out the wire 
rope, though for distances of from 300 to 1,500 feet it is consid- 
ered a cheap and simple method of transmitting power. Two 
systems are used, namely, the continuous or endless rope used in 
operating cableways, haulage systems and tramways, and the 
single loop, the latter being simply a modification of belt driving. 

161. Single Loop System. — To transmit power by means of a 
single loop with a minimum amount of slippage, a certain amount 
of pressure between the surfaces in contact is necessary. This 
pressure depends upon the weight and the tension of the rope. 
Therefore, for short spans it is frequently necessary to use a large 
rope in order to get the proper weight, although the tension may 
be increased by resplicing or by the introduction of a tightener. 
The last two methods are not considered good practice as the 
rope may be strained too much, and in addition, the filling in the 
bottom of the grooves of the sheaves wears away too rapidly. 

Experience has shown that transmitting power by means of 
wire rope is generally not satisfactory when the span is less than 
50 to 60 feet. This is due to the fact that the weight of the rope 
is not sufficient to give the requisite friction without the use of 
tighteners. When the distance between the shaft centers ex- 
ceeds 400 feet, successive loops are used; that is, the driving 
sheave of the second loop is keyed fast to the shaft of the driven 
sheave of the first loop, or double-groove sheaves may be used. 

162. Wire Transmission Rope. — Wire rope used for transmit- 
ting power consists of six strands laid around a hemp or wire core, 
each strand containing seven wires. The rope with a hemp core 
is more pliable and for that reason is generally preferred for power 
transmission . As mentioned in a preceding paragraph, large ropes 
are occasionally required to get a satisfactory drive, and in such 



Art. 163] 



STRESSES IN WIRE ROPE 



213 



installations a six-strand nine teen-wire rope is to be preferred. 
The 6X7 construction of rope, having much larger wires, will 
stand more wear than the 6X19 construction, but requires much 
larger sheaves. The material used in the manufacture of wire 
transmission rope is iron, crucible cast steel, and plow steel. In 
Table 46 is given information pertaining to two kinds of high- 
grade transmission rope. 

163. Transmission Sheaves. — The sheaves for transmission 
rope are quite different from those used with manila rope, as will 
be seen by consulting Fig. 86. The grooves are made V shape 
with a space below, which is filled with leather, 

rubber, or hardwood blocks. One prominent 
manufacturer uses alternate layers of leather 
and blocks of rubber for a filling. The func- 
tion of this filling is to increase the friction be- 
tween the rope and sheave and at the same 
time reduce the wear of the rope to a mini- 
mum. The filling should have a depression 
so that the rope will run central and not come 
into contact with the iron sides of the grooves. 
The speed of the rim of the sheave should not 
exceed 5,000 feet per minute. 

The diameter of the sheave should be made 
as large as practicable consistent with the per- 
missible rim speed. 

Large sheaves decrease the bending stresses FlG - 86 - 

and at the same time increase the transmitting 
power of the rope. In Table 46 are given the minimum diam- 
eter of sheaves that should be used with the various sizes of 6 X 
7 and 6 X 19 transmission rope. 

164. Stresses in Wire Rope. — The maximum stress in a wire 
rope due to the power transmitted should always be less than the 
difference between the maximum allowable stress and that due 
to the bending of the rope. For the magnitude of the bending 
stress in a rope running over a sheave, consult Fig. 78. As the 
bending stress decreases, the load stress may be increased, but 
the sum of these two separate stresses should never exceed from 
one-third to two-fifths of the ultimate strength of the rope given 
in Table 46. No provision is made, however, for the weakening 
effect of a splice in the rope. To prevent slippage between the 




214 



SAG OF WIRE ROPE 



[Chap. IX 



rope and the sheave, the ratio of the tight to the loose tension 
must have a value given by the following expression, which may 
be derived directly from (218) by making the angle /3 equal to 
90 degrees. The symbols used have the same meaning as as- 
signed to them in Art. 144. 

T _ wv* 

— 2 = e*» (254) 

T wv 2 

Q 
For the coefficient of friction /x, Mr. Hewitt in his treatise pub- 
lished by the Trenton Iron Co., recommends the values given in 
Table 50. 



Table 50. — Coefficients of Friction fob Wiee Rope 



Type of groove 


Condition of rope 


Dry 


Wet 


Greasy 


Plain groove 

Wood-filled 


'0.170 
0.235 
0.495 


0.085 
0.170 
0.400 


0.070 
140 




205 







To determine the horse power capable of being transmitted by 
a given size of wire rope use (221), substituting for// in that equa- 
tion the proper value from Table 50. As in the case of manila 
ropes, there is a speed that makes the horse power transmitted 
a maximum and beyond which the horse power decreases. To 
determine the speed corresponding to the maximum horse power, 
use (222). 

165. Sag of Wire Rope. — The question of sag was discussed in 
Art. 147 in connection with manila ropes and the various formu- 
las deduced also apply in the present discussion. It is desirable, 
whenever possible, to make the lower rope of a transmission do 
the driving; the upper or slack rope sags, thereby increasing the 
angle of contact on both sheaves and, at the same time, the trans- 
mitting capacity of the installation. According to the Trenton 
Iron Co., the sag of the tight or lower rope should be about one- 
fiftieth of the span, and that of the slack rope about double this 
amount. 



Art. 165] REFERENCES 215 



References 

The Constructor, by F. Retjleaux. 

Machine Design, Construction and Drawing, by H. J. Spooner. 

Elements of Machine Design, by W. C. Unwin. 

Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 

Die Drahtseile, by J. Hrabak. 

The Application of Wire Rope to Transportation, Power Transmission, 
etc., by W. Hewitt. 

Wire Rope Handbook, by American Steel and Wire Co. 
The Transmission of Power by Wire Rope, Mine and Minerals, April 
1904. 



CHAPTER X 
CHAINS AND SPROCKETS 

The various types of chains found in engineering practice may, 
according to their use, be grouped into the following classes : 
(a) Chains intended primarily for hoisting loads. 
(6) Chains used for conveying as well as elevating loads, 
(c) Chains used for transmitting power. 

HOISTING CHAIN 

166. Coil Chain. — The kind of chain used on hoists, cranes, 
and dredges is shown in Fig. 87(a) and is known as coil chain. 







— P — 






* 


( r 


* 


5 


T"#K~7"- 


> 


1 


v L_^-y- 


.1 _^_0 


^-H 




— : I "— 





(a) 




The links are made either of an, elliptical shape or with the sides 
parallel, and the material used should be a high-grade refined 
wrought iron or an open-hearth basic steel. A chain made of the 
latter material has a higher tensile strength and at the same time 
stands greater abrasive wear than one made of wrought iron. In 
order to insure flexibility in a chain, the links should be made 

216 



Art. 167] 



HOISTING CHAINS 



217 



small. A small link has the added advantage that the bending 
action at the middle and at the end of the link due to the pull 
between adjacent links is decreased. In Table 51 are given the 
general dimensions of the link, weight per foot, and the approxi- 
mate breaking, strength of the commercial sizes of dredge and 
crane chains. 







Table 


51. — Hoisting Chains 




Size 


Pitch 


Weight 
per foot 


Outside 

length, 

in. 


Outside 

width, 

in. 


Dredge and 
shovel 


BBB crane 


















Approx. breaking load, lb. 


H 


2 %2 


0.75 


lMe 


Vs 


5,000 


• 4,000 


Me 


2 %2 


1.00 


iy 2 


We 


7,000 


6,000 


% 


% 


1.50 


i% 


1M 


10,000 


9,000 


Vie 


l%2 


2.00 


2Ke 


1% 


14,000 


13,000 


y 2 


1% 


2.50 


2H 


1% 


18,000 


17,000 


He 


1^2 


3.25 


2 5 A 


1% 


22,000 


20,000 


% 


1 2 %2 


4.00 


3 


2He 


27,000 


26,000 


x Ke 


1 X K6 


5.00 


3M 


2M 


32,500 


30,000 


H 


1% 


6.25 


SH 


2H 


40,000 


36,000 


13 Ae 


2Kb 


7.00 


m 


2% 


42,000 


40,000 


% 


2Ke 


8.00 


4 


2% 


48,000 


44,000 


x He 


2Ke 


9.00 


m 


3Ke 


54,000 


50,000 


1 


2M 


10.00 


4% 


3M 


61,000 


57,000 


IHe 


2% 


12.00 


4% ■ 


3^6 


69,000 


65,000 


1H 


2% 


13.00 


5^ 


3% 


78,000 


72,000 


lMe 


3Ke 


14.50 


5%6 


3% 


88,000 


80,000 


IX 


3H 


16.00 


5% 


4^ 


95,000 


88,000 


1Kb 


3% 


17.50 


Qy 8 


4M 


104,000 


96,000 


1% 


3%6 


19.00 


6Ke 


4Kb 


114,000 


104,000 


IHe 


3% 


21.16 


6% 


4% 


122,000 


116,000 


m 


3% 


23.00 


7 


5 


134,000 


124,000 


^Ae 


4 


25.00 


7V 8 


5%6 


142,000 


132,000 


1% 


4^ 


28.00 


7M 


5K 


154,000 


144,000 


m 


m 


31.00 


8M 


5% 


166,000 




i% 


5M 


35.00 


9^ 


6% 


190,000 




2 


5% 


40.00 


10 


6% 


216,000 




2M 


6% 


53.00 


HM 


7% 


273,000 




2^ 


7 


65.00 


12K 


8% 


337,000 




2% 


7M 


73.00 


13 


9^ 


387,000 




3 


7% 


86.00 


14 


9% 


436,000 





167. Stud-link Chain. — A type of chain known as the stud-link 
chain is shown in Fig. 87(6). It is used mainly in marine work 



218 



CHAIN DRUMS 



[Chap. X 



in connection with anchors and moorings. The chief advantage 
of the stud-link chain is that it will not kink nor entangle as 
readily as a coil chain. Experiments show that for the same size 
of link, the addition of the stud results in a decrease of the ulti- 
mate strength of the chain. An analysis of the stresses in chains 
shows, however, that within the elastic limit the stud-link chain 
will carry a much greater load than the open-link chain. See 
Bulletin No. 18 Univ. of Illinois Experiment Station, G. A. Good- 
enough and L. E. Moore. 

168. Chain Drums and Anchors. — In practically all cases 
where short-link chains are used for heavy service, as on cranes 
and dredges, drums are used for winding up the chain. Such 




Fig. 88. 



drums should always be provided with machined grooves. Two 
forms of such grooves are shown in Fig. 88, and the dimensions 
given in Table 52 will be found convenient for layout purposes. 





Table 52.— 


Dimensions of Grooves for Chain Drums 




Type 


Di- 
men- 
sion 


Size of chain 


n 


Vie 


H 


He 


H 


% 


% 


13 Ae 


7 A 


me 


1 


(a) 


a 
b 


IK 

H 


1 X M6 

%2 


He 


2Ke 


2Kb 


2V 2 

1S A2 


2 l Xe 
He 


2% 

1 H2 


sy 8 

V2 


SHe 


3K 
He 


(b) 


a 
b 


He 


VAe 

Hi 


lHe 


m 

%2 


He 


2He 

X YZ2 


2Hs 

H 


2% 

13 A2 


2Y2 

He 


2"He 

1S /Z2 


2% 

V2 



The diameter of the drum depends upon the speed of hoisting, 
the loads to be raised, and the life of the chain. For close-link 
chain, it has been found by experience that the drum diameter 
should not be made less than twenty times the thickness of the 



Akt. 168] 



CHAIN ANCHORS 



219 



chain material, and it is better to make it about thirty times the 
thickness of material. If the drum is made small in diameter 
relative to the size of the chain, the bending action on the link 




referred to in Art. 166, will be excessive, thus decreasing the life 
of the chain. 

The drum should always be made of sufficient length so that 
the required length of chain may be wound upon it in a single 
layer. It is considered good design to have one or two coils of 



220 



CHAIN ANCHORS 



[Chap. X 



chain remaining on the drum when the load is in its lowest posi- 
tion, thus reducing the stress coming upon the anchor. The cor- 
rect stress analysis for a hoisting drum is rather complex, and the 
approximate method outlined in Art. 158 for a drum using wire 
rope is applicable to chain drums. 

Anchors. — The method of anchoring the free end of the 
chain to the drum should be given attention. In Fig. 89 are 
shown three designs taken from the practice of several crane 
builders. The method shown in Fig. 89(a) is faulty for the fol- 
lowing reasons: (1) The hole for the tongue of the anchor is 




Fig. 90. 



drilled in the chain groove, thus increasing the bending action on 
the tongue. (2) In case the chain should ever assume the posi- 
tion indicated by the dotted lines, the cap screw a will receive the 
greatest load instead of the tongue of the anchor. 

The design shown in Fig. 89(6) overcomes the first objection 
in that the tongue of the anchor is placed in a hole drilled into 
the solid metal. The second objection, however, also applies to 
this design. In Fig. 89(c) is shown a construction that is cheap 
to make and at the same time overcomes both objections. In 
certain designs of drums it may not always be as convenient to 
attach an anchor of this type as one of the first two types. 



Art. 169] 



CHAIN SHEAVES 



221 



169. Chain Sheaves. — Sheaves are of two classes, namely those 
that merely guide the chain as in changing direction, and those 
that are fitted with pockets to receive the links of the chain. 
The latter class is used extensively in chain hoists in place of 
drums, also for transmitting power under certain conditions. 

(a) Plain sheaves. — Designs of plain sheaves, referred to as the 
first class, are shown in Figs. 90 and 91. The proportions of the 
two types of sheaves are given in Table 53. For sheaves of large 
diameters, arms are used, while with small sheaves the web cen- 
ter has given better satisfaction. The web centers may be plain, 




Fig. 91. 
Table 53. — Dimensions op Plain Chain Sheaves 



Size of 
chain 



Type— Fig. 



Type— Fig. 91 



IK 

m 



1 % 

M 

% 
i 



9 i6 



me 

l 15 Ae 
2% 

3M 
3M 



1%6 

l 13 /u 
2Ke 



VZ2 

He 



XI 6 

y 2 



2 

2Ke 
2M 
3^ 



222 



CHAIN SHEAVES 



[Chap. X 



or if it is desirable to decrease the weight, round holes may be 
cut out as shown in Figs. 90 and 91. To give stiffness to the 
center web, the side ribs as shown in Fig. 90 are added. 

(b) Pocket sheaves. — Sheaves similar to the one shown in Fig. 
92 are called pocket sheaves and are used principally on chain 
hoists in place of drums. The horizontal links fit into pockets 
cast in the periphery of the sheave, while the vertical links fall 





~""i e r" 

1 


z^r.T]f7i^ 


1 


W 


ifi 


1 


o 


rl 




E 












o 

4- 

a. 


$2 




n 


iiiiisi 













Fig. 92. 



into a central groove as shown. In order to design a sheave of 
this type, the various calculations involving the formulas given 
below must be carried out with considerable accuracy. The 
dimensions of the chain and the number of pockets or teeth T 
desired enable one to derive a formula for the so-called pitch 
diameter D. 



Art. 169] CHAIN SHEAVES 223 

From Table 51, the dimensions a and b of Fig. 92 for the chosen 
size of chain are established, and from the number of pockets T, 
we find for the angle a, 

180 ° (oxe\ 

a = —^- (255) 

From the geometry of the figure, the following equations are 
obtained: 

sin (« - 0) = £ (256) 

sin/3=^ (257) 

Eliminating D and solving for /3, we obtain 

tan/3 = Sma (258) 

cos a + r 
o 

Since the dimensions a and 6 as well as the angle a are known in 
any given case, (258) is used to determine the angle |8; having 
determined this angle, the pitch diameter D of the sheave is found 
by means of (257). 

The rim of the pocket sheave may be proportioned by the fol- 
lowing empirical formulas, in which d and w denote the dimen- 
sions of the chain links as given in Fig. 87. Referring to Fig. 92 : 

c = d + (%" to M 6 ") 
e = %d 

f = w+ {y s " to Y lQ ") 

The thickness of the web should not be made less than the 
diameter of the material in the chain, and the diameter of the 
hub should be approximately twice the diameter of the pin sup- 
porting the sheave. 

Having determined the pitch diameter and the general pro- 
portions of the rim, web, and hub of the sheaves, the next step 
is to make a full-size drawing with the pockets from }i to }i inch 
longer than the link. The layout of the tooth shown in Fig. 92 
represents the tooth form at the center line of the sheave and 
not at the side of the central groove where it should begin. How- 
ever, since the pockets are to be made somewhat longer than the 
links, the tooth may be given this form at the side of the central 



(259) 



224 



RELATION BETWEEN P AND 



[Chap. X 



groove and it will be found that sufficient clearance is thus pro- 
vided in the majority of cases. The face of the tooth, or that 
part lying above the pitch circle, may be drawn with a radius 
equal to three or four times the diameter of the chain material. 

170. Relation between P and Q. — When a chain is wound on, 
or unwound from, a sheave, the relative motion between the links 
on the running-on and -off sides introduces frictional resistances. 
The turning of one link in another is similar to that of a journal 
running in its bearing, and the relation between the applied effort 
P and the resistance Q will be based upon the theory of journal 
friction. 

In Fig. 93 is shown diagrammatically a chain raising a load Q 
by means of an effort P. 




Let D = pitch diameter of the sheave. 
-Do = diameter of the sheave pin. 
d = size of the chain. 
n = coefficient of journal friction. 
fjL c = coefficient of chain friction. 

On the load side, the link a in moving from the position E to 
that at D turns through the angle a relative to the link b. To 
overcome the frictional resistance between these links requires an 

amount of work to be done by the effort P equivalent to fj. c Q-^a. 

At the same time that the link a is moving from E to D, a link 
on the effort side is running off, the frictional resistance of which 

requires work to be done equivalent to HcP^a. In addition to 



Art. 171] CHAIN BLOCK 225 

these resistances, the friction of the sheave pin must be overcome. 
For the loading shown in Fig. 93, the pressure on the pin is 
(P + Q) J therefore, the work required to overcome the friction 
of the pin for an angular displacement a of the sheave is equiva- 
lent to na (p + Q) ^°- 

The useful work done is 9 ; hence, the total work required 
by the effort P to raise the load Q is 

^ = «^ + , a (p + <3 )f + ^(P + Q)| (260) 

from which 

*- &t£t'S \e~Bi ^6!) 

The value of K varies from 1.04 to 1.10, the first value applying 
to lubricated chains and the latter to chains running dry. 

Efficiency of chain sheave. — By applying the definition of 
efficiency to the case discussed above, we find that the efficiency 
is 

1 



V 



K 



(262) 



Introducing the values of K given above, it follows that the 
efficiency of a chain sheave having the chain lubricated is 96 per 
cent., while the same sheave with the chain running dry has an 
efficiency of approximately 91 per cent. 

171. Analysis of a Chain Block. — With the aid of the principle 
discussed in Art. 170, it is a simple matter to analyze blocks reefed 
with chains. As an example, it is required to determine the mag- 
nitude of the effort P that is required to raise a load Q by means 
of a differential chain block, similar to the one shown diagram- 
matically in Fig. 94. As indicated in the figure, an endless chain 
is reefed around the compound sheave ab and the lower sheave c. 
As constructed, this block is always made self-locking, except 
occasionally when the chain becomes very greasy, the load will 
run down. Due to its self-locking property, the efficiency is 
rather low. 

(a) Raising the load. — During one revolution of the compound 
sheave, the part d of the chain rises a distance 2 irR, while the 
part e descends a distance 2rr; hence the sheave c and the load 



226 



CHAIN BLOCK 



[Chap. X 



Q rise a distance ir(R — r). 
effort is 2tPoR; hence 



Without friction, the work of the 



from which 



2 tP R = irQ(R 
Q 



r), 



Po 



(1-n), 



(263) 



in which n denotes the ratio „' Evidently any desired reduc- 
tion may be obtained by varying 
the difference (R — r). 

Considering the lower sheave 
c, it is evident from (261) that 
the relation between the tensions 
in the running-off and running-on. 
chains is 7\ = K\T 2 , where K\ 
depends upon the size of the chain 
and the diameter of the sheave 
c. Furthermore, Q = T x + T 2 ; 
hence, the following expressions 
are obtained : 




T, = 



l + tfi 
Q 

l + Xi 



(264) 



Now the compound sheave ab 
is held in equilibrium by the 
forces P, 7\, T 2 , the pin reaction, 
and the friction forces. Taking 
FlG - 94 - account of friction, we then have 

PR + TV = K 2 TrR (265) 

where K 2 depends upon the size of the chain and the diameter of 
the sheave a. 

Combining (264) and (265), the magnitude of the effort be- 
comes 

K X K, 



*-mir\* 



(266) 



Replacing K x and K 2 by an average value denoted by K, 
(266) becomes 



Art. 171] CHAIN BLOCK 227 

The efficiency for the differential chain hoist is given by the 
expression 

(b) Lowering the load. — When the load is lowered, the frictional 
resistances all act in the opposite sense, and the analysis is given 
by the following equations: 



T Q 



l + K 
KQ 



To = 
- l + K 

T X R = K(P)R + KT 2 r 



(269) 



in which (P) represents the pull required on the chain so as to 
prevent running down of the load. Combining the three equa- 
tions given in (269), we obtain the following expressions for the 
effort (P) and the efficiency (rj) for reversed motion: 

« - iT+f ] ^ 

W - M di^j q] (271) 

(c) Conditions for self-locking. — Whether the hoist shown in 
Fig. 94 is self-locking or not depends upon the values of K and 
n. For self-locking, it is apparent that (P) <0; hence the crit- 
ical value of n at which the self-locking property commences is 
given by the equation, 

from which it follows that 

1 - nK 2 < (272) 

. r . 
Therefore, the critical value of the ratio ^ is 

K 

(273) 
For a self-locking hoist, n > -^ . (274) 

(d) Experimental data. — An investigation of six sizes of differ- 
ential chain blocks having capacities from 500 to 6,000 pounds, 
inclusive, gave actual efficiencies varying from 28 per cent, for the 
larger capacities to 38 per cent, for the smaller sizes. Further- 



n 


— 


K* 


n 


> 


1 
K 2 



228 



DETACHABLE CHAIN 



[Chap. X 



more, it was found that the value of K as determined from equa- 
tion (267) or (268) varied from 1.054 to 1.09. 

CONVEYOR CHAINS 

For the purpose of conveying and elevating all kinds of mate- 
rial, various types of chains are used. These chains may be 
adapted very readily to a wide range of conditions by using spe- 
cial attachments, such as buckets and nights. The chains used 
for this class of service may in general be grouped into the follow- 
ing two classes: (a) detachable or hook-joint, and (6) closed- 
joint. 

172. Detachable Chain. — The detachable, or hook-joint chain 
shown in Fig. 95 is used very extensively, and under favorable 
conditions gives good service. The chain shown is made of 




malleable iron; but there is a form of hook chain now obtainable 
that is made of steel. Since the joints between the links are of 
the hook or open type, this kind of chain is not well adapted to 
the elevating and conveying of gritty bulk material; however, if 
the joints are properly protected, slightly gritty material may be 
handled. In addition to this class of service, hook-joint chains 
are frequently used for power-transmission purposes at moderate 
speeds, say not to exceed 600 feet per minute for the Ewart chain 
shown in Fig. 95 and a considerably higher figure for the lock 
steel chain. For elevating and conveyor service, the speeds 
seldom exceed 200 feet per minute. 



Art. 173] 



TABLE OF EWART CHAINS 



229 



173. Strength of Detachable Chain. — In Table 54 is given 
general information pertaining to the standard sizes of Ewart 
detachable chain, manufactured by the Link Belt Co. In addi- 
tion to the sizes listed, a large number of special sizes are made. 
In order that a chain drive may be durable, a proper working load 





Table 54- 


-Ewart Detac 


hable Chain 




Chain No. 


Approx. links 
per ft. 


Aver, pitch 


Weight 
per ft. 


Ultimate 

strength, 

lb. 


25 


13.30 


0.902 


0.239 


700 


32 


10.40 


1.154 


0.333 


1,100 


33 


8.60 


1.394 


0.344 


1,190 


34 


8.60 


1.398 


0.387 


1,300 


35 


7.40 


1.630 


0.370 


1,200 


42 


8.80 


1.375 


0.570 


1,500 


45 


7.40 


1.630 


0.518 


1,600 


51 


10.40 


1.155 


0.707 


1,900 


52 


8.00 


1.506 


0.848 


2,300 


55 


7.40 


1.631 


0.740 


2,200 


57 


5.20 


2.308 


0.832 


2,800 . 


62 


7.30 


1.654 


1.022 


3,100 


66 


6.00 


2.013 


1.158 


2,600 


67 


5.20 


2.308 


1.196 


3,300 


75 


4.60 


2.609 


1.311 


4,000 


77 


5.20 


2.293 


1.456 


3,600 


78 


4.60 


2.609 


1.909 


4,900 


83 


3.00 


4.000 


1.944 


4,950 


85 


3.00 


4.000 


2.400 


7,600 


88 


4.60 


2.609 


2.438 


5,750 


93 


3.00 


4.033 


2.670 


7,500 


95 


3.00 


3.967 


3.000 


8,700 


103 


3.90 


3.075 


4.087 


9,600 


108 


2.55 


4.720 


3.570 


9,900 


110 


2.55 


4.720 


4.437 


12,700 


114 


3.70 


3.250 


5.180 


11,000 


122 


2.00 


6.050 


7.000 


15,000 


124 


3.00 


4.063 


6.666 


12,700 


146 


2.00 


6.150 


6.240 


14,400 



must be used. This depends upon the speed and the class of 
service for which the chain is used. After a considerable number 
of years of experimental work, the Link Belt Co. has established a 
series of factors that may be used for arriving at the proper work- 
ing stresses at various speeds. In Fig. 96, the factors just re- 
ferred to have been plotted so as to bring them into more con- 



230 



CLOSED-JOINT CHAINS 



[Chap. X 



venient form for general use. To determine the working stress 
for any particular size of chain, multiply the ultimate strength as 
given in Table 54 by the speed coefficient obtained from the graph 
in Fig. 96. 

174. Closed-joint Chains. — As the name implies, this type of 
chain has a closed joint; because of this fact it is well adapted to 
the elevating and conveying of gritty and bulk material, as well 



0.06 






TfF= 



0.14'G 
I 



0.I5-, 



0.17 



500 


400 300 200 


Speed of 


Chain - ft. per min. 




Fig. 96. 



as transmitting power at moderate speeds. A large number of 
different types of closed-joint chain are now manufactured. In 
Figs. 97, 98 and 99 are shown three types, the first two being 
made of malleable iron and the third of steel. The closed-joint 
chains are made in the same sizes as the detachable chains; hence 
the sprockets are interchangeable. In the better grades of closed- 
joint chains, the pins and bushings used are frequently made of 
hardened steel. 



Art. 174] 



CLOSED-JOINT CHAINS 



231 




Fig. 97. 




Fig. 98. 




232 



STRENGTH OF CLOSED-JOINT CHAINS [Chap. X 



175. Strength of Closed-joint Chain. — The information given 
in Table 55 pertains to the chains shown in Figs. 97 and 98. The 
chain shown in Fig. 97 is manufactured by the Link Belt Co. and 
is known as the "400 Class Closed-end Pintle Chain." To 
determine the proper working load for this chain, the ultimate 
strength given in Table 55 must be multiplied by the so-called 
speed coefficient mentioned in Art. 173, values of which may 
be obtained from Fig. 96. 





Table 55 


. — Closed-joint Conveyor and Power Chains 




Link Belt Co's. "400" Class 


Jeffrey-Mey-Obern type 


„ . 


_a 




s 


- ft 


■a 




M ■ 




£ 


g. 


jO 


6 
S3 

'3 

6 


a 1 " 
ftfe 
< p,- 


'3. 

> 


ft 


3 8 a 
S&1 


a - 

a "5 

.a a 


6 
.5 

6 


Is 


a 

"ft 

> 

< 


ft 

.1 
1 


gft . 
3£ a 

|51 

5! ft 


l-a 
■43 £ 


434 


8.6 


1.398 






3,600 


25 


13.30 


0.903 


0.416 




1,000 


442 


8.8 


1.375 


1.470 




5,900 


33 


8.60 


1.395 


0.525 




2,200 


445 


7.4 


1.630 


1.428 




5,900 


34 


8.60 


1.395 


0.758 


700 


2,650 


445H 


7.4 


1.630 






3,600 


42 


8.75 


1.370 


1.090 




3,000. 


447 


7.4 


1.630 






5,900 


45 


7.40 


1.623 


1.090 




3,300 


452 


8.0 


1.506 






7,000 


50 


12.00 


1.000 


0.548 




1,900 


455 
462 


7.4 
7.3 


1.630 
1.634 


1.783 
2.314 




7,300 
9,000 
















52 


8.00 


1.517 


1.480 




4,750 


467 


5.2 


2.308 


1.336 




6,200 


55 


7.40 


1.630 


1.400 




4,925 


4,072 


7.3 


1.654 


2.445 




9,000 


57 


5.20 


2.307 


1.460 




5,800 


477 


5.2 


2.293 


1.960 




10,000 


62 


7.30 


1.647 


1.920 




5,850 


X477 


5.2 


2.308 


1.825 




7,300 


67 


5.20 


2.308 


1.680 


600 


6,000 


483 


3.0 


4.000 






15,000 


75 


4.60 


2.619 


1.970 




7,350 


488 


4.6 


2.609 


2.769 




12,000 


77 


5.20 


2.311 


1.740 




6,500 


488^ 


4.6 


2.609 






13,000 


77H 


5.20 


2.311 


2.320 




8,300 


4,103 


3.9 


3.075 


5.398 




20,000 


























4,124 


3.0 


4.100 






33,000 


78 

78K 

83 


4.60 
4.60 
3.00 


2.620 
2.620 
3.970 


2.170 
2.880 
3.100 




8,050 
9,900 
11,425 














85 


3.00 


3.960 


3.950 


500 


13,500 














88 


4.60 


2.610 


2.630 




8,300 














103 
108 


4.00 


3.058 


5.290 




13,530 




2.55 


4.751 


5.180 


400 


14,800 














121 


2.06 


6.042 


3.600 


500 


16,600 














122 


2.00 


6.109 


8.510 


300 


24,980 














124 


3.00 


4.074 


11.770 


300 


40,000 














146 


2.00 


6.215 


8.120 


300 


23,500 



The chain shown in Fig. 98 is manufactured by The Jeffrey 
Mfg. Co. and is known as the "Mey-Obern" type. The proper 
working stress for any particular speed may be found by using 
the speed coefficients given in Fig. 96. 

The chain shown in Fig. 99 differs from those shown in Figs. 



Art. 175] 



TABLE OF UNION STEEL CHAINS 



233 



97 and 98 in that the body of the link is stamped and formed 
from one piece of steel, the sides being connected across the top 
by a bridge as shown. This chain is manufactured by The Union 
Chain and Mfg. Co. of Seville, Ohio, and is made in two types, 
namely, the bushing type and the roller type. The information 
contained in Table 56 pertains to the roller type shown in Fig. 
99, the upper part of the table showing the commercial sizes used 
mainly for power transmission, while the lower part gives the sizes 



Table 56. — Union Steel Chains 



Chain 
No. 



3R 



4R 



5R 



6R 



m 



l. 1 : 



Length 



Ultimate 
strength, lb. 



3,500 



5,000 



7,500 



10,500 



7R 



8R 



\% 



1 

IK 



14,000 



18,000 











Rollers 






Chain 


links 
per ft. 


Average 






Ultimate 




No. 


pitch 






strength, lb. 








Length 


Diameter 






14 


8.0 


1.50 


M 




8,000 





15R 


7.4 


1.62 


% 


l He 


6,000 


M 


16R 


6.0 


2.00 


1 


1H 


12,000 




17R 


5.2 


2.31 


1 


m 


10,000 




18R 


4.6 


2.61 


1M 


% 


12,000 





19 


3.9 


3.07 


1^16 


m 


15,000 




21 


3.4 


3.51 


m 


i% 


22,000 




22 


3.0 


4.00 


iy 2 


m 


30,000 




30 


2.0 


6.00 


i% 


i% 


40,000 



234 



CHAIN SPROCKETS 



[Chap. X 



that have been designed to run on standard sprockets used for 
detachable chain. The latter type of chain is used for either 
power or conveyor service. To arrive at the working stress for 
a given speed, multiply the ultimate strength given in Table 56 
by the speed coefficient taken from the graph in Fig. 96. 

176. Sprockets for Detachable Chains. — Cast sprockets are 
generally inaccurate due to shrinkage and rapping of the pattern; 
hence in order to get satisfactory service they should be made a 




Fig. 100. 



trifle large and then ground to fit the chain. The sprockets made 
from ordinary cast iron give good service, especially if both the 
chain and the face of the sprockets are lubricated with a heavy 
oil or a thin grease. For severe service manufacturers furnish 
sprockets having chilled rims and teeth, while the hubs are soft 
for machining purposes. 

Armor-clad sprocket. — Another form of sprocket that is in- 
tended to give great durability is shown in Fig. 100. It consists 
of a cast-iron central body in the periphery of which are milled 
slots. Into these slots are fitted the teeth a, which are formed 



Art. 177] 



CHAIN SPROCKETS 



235 



from special steel strips. To fasten the teeth rigidly in the body, 
the ends are expanded by means of a steel pin b, and lateral dis- 
placement is prevented by washers and riveting. The teeth are 
heat treated and may be removed very readily. This design of 
sprocket is used by The Union Chain and Mfg. Co. 

It is claimed that these sprockets, and also sprockets having 
chilled rims and teeth, are more economical since they last con- 
siderably longer, although they cost approximately 50 per cent, 
more than the gray iron sprockets. 

177. Relation between Driving and Driven Sprockets. — Theo- 
retically the pitch of the sprocket teeth and that of the chain 
should be exactly the same; but as chains may vary a trifle from 





Fig. 101. 



the exact pitch, and as the wear of the joints tends to lengthen 
the pitch, some provision must be made to take care of this elon- 
gation or the chain will ride on the teeth of the sprocket. To 
overcome this riding action, the teeth of sprockets are given back 
clearance; that is, their thickness on the pitch circle is made less 
than the dimension s shown in Figs. 95, 97, 98, and 99. Further- 
more, the pitch of the teeth is increased or decreased, depending 
upon whether the sprocket is the driving or driven member of 
the transmission. 

In Fig. 101 are shown two sprockets transmitting power, a 
being the driver and b the driven. This figure shows the correct 



236 SPROCKET TOOTH FORM [Chap. X 

chain action, and it should be noted that on each sprocket the 
entire load transmitted by the chain comes upon one tooth, 
namely upon the one at the point where the links run off the 
sprocket. Referring to Fig. 101, it is evident that the loaded 
tooth on the driving sprocket in pushing the chain forward per- 
mits the disengaging link to roll out to the tip of this tooth and 
at the same time the chain creeps backward a distance equal to 
the increment x. By the time the driving tooth is completely 
disengaged, the following tooth of the wheel is seated firmly 
against the following link; hence it follows that the chordal pitch 
Pi of the sprocket is greater than the pitch of the chain. A simi- 
lar analysis of the action of the chain on the driven sprocket shows 
that the disengaging link in rolling out on the loaded tooth creeps 
ahead a distance equal to the increment y, thus bringing the follow- 
ing link and tooth into intimate contact. It is evident, therefore, 
that the chordal pitch p 2 of the sprocket b should be less than the 
pitch of the chain. The condition may also be met by making 
the chordal pitch p 2 of anew sprocket equal to the chain pitch, and 
as soon as the wear appears the links creep away from the teeth 
producing the action just discussed. 

Sprockets laid out as shown in Fig. 101 are likely to show ex- 
cessive wear since one tooth must carry the entire load trans- 
mitted by the chain. According to information furnished by 
The Jeffrey Mfg. Co., the amount that the driving sprocket is 
made larger than the theoretical size depends upon the pitch of 
the chain, the size of roller or hook of the link, the strength of the 
chain, and the number of the teeth in the sprocket. 

178. Tooth Form. — From the discussion given in Art. 177, it is 

evident that the teeth of sprockets must be given considerable 

clearance so as to permit the chain 

Table 57.-Sprocket Teeth to elongate due to the load as well 

Factors ,-, ,, . , , 

as the wear on the pins and not per- 



8 to 12 
13 to 20 
21 to 35 
36 to 60 



Factor mit it to ride on the flanks of the 

teeth. If the chain transmission 

. 75 to . 80 is designed properly, each tooth 

°- 70 . comes into action only once per 

55 t 60 revolution of the sprocket; hence, 

" in sprockets having large numbers 

of teeth, the wear on the tooth 
flanks is distributed over more teeth, and for that reason the 
thickness of the tooth at the pitch line may be made less than 



Art. 178] 



SPROCKET TOOTH FORM 



237 



in smaller sprockets. The data included in Table 57 will 
serve as a guide in laying out the teeth of sprockets. To ob- 
tain t, the thickness of the tooth at the pitch line, for any given 
size of chain multiply the length of the available tooth space in 
the link by the factors given in the table. These factors represent 
the practice of the Link Belt Co. and are based upon experience 
with chains in service. By the available tooth space in the 
link is meant the dimension s in Figs. 95, 97, and 98. 

Having decided upon the size of chain and the number of teeth 
in the sprocket for the particular case under consideration, deter- 






u> 




— 1 











Fig. 102. 



mine the pitch diameter of the sprocket by the following ex- 
pression : 

V 

sin a 



D 



(275) 



in which D denotes the pitch diameter, p the pitch of the chain, 
and a equals 180 degrees divided by the number of teeth in the 
sprocket. 

Having calculated the pitch diameter, the sprocket teeth may 
be laid out as shown in Fig. 102. The root circle diameter, as 
shown in the figure, is fixed by the dimensions of the link. An 
examination of a considerable number of sprockets made by lead- 
ing manufacturers seems to indicate that the outline of the tooth 
may be made a straight line between the root circle and the 
rounded corner at the top. The radius r of this corner varies 
from }{q inch for small chains to about % inch for the larger 



c 


= 


0.5 w 












( x Mfi w 


for small chains 


a 


= 


\ w - 


-h 


" for large 


chains 


e 


= 


% w 








f 


= 


%w 








9 


= 


0.7 w 








k 


= 


1.25 


(P 


-*) 


. 



238 PROPORTIONS OF SPROCKETS [Chap. X 

chains. The inclination of this line must provide sufficient 
clearance to prevent interference between the tooth and the link 
when the latter is entering or leaving the sprocket. The flank 
of the tooth is joined at the root circle by a fillet having a radius 
less than that of the hook of the link. 

179. Rim, Tooth, and Arm Proportions. — (a) Rim and tooth. — 
The rim of the sprocket may be proportioned in a general way by 
the following empirical formulas taken from Halsey's Handbook 
for Machine Designers and Draftsmen. In these formulas the 
dimensions denoted by p and w are obtained from the size of the 
chain under consideration. 



(276) 



(b) Arm proportions. — Sprockets are made with a web center 
or with arms. For very small pitch diameters, solid web centers 
having a thickness determined by the dimension a in Fig. 102 
should be used. For larger diameters up to, say, 12 or 15 inches, 
web centers with holes may be used; but in these cases the web 
thickness should be made equal to approximately six-tenths 
of the dimension a as determined by means of (276). For diam- 
eters exceeding 12 or 15 inches, the sprockets should be con- 
structed with arms, the dimensions of which may be obtained 
by the following analysis : 

Let W = breaking load of the chain. 

S = permissible working stress for the material. 
b = thickness of the arm at the center of the 

shaft. 
h = depth of the arm at the center of the shaft. 
n = number of arms, 4 to 6. 

To be on the safe side, the arm of the sprocket is designed for a 
load exceeding that coming upon the chain. This condition is 
met by assuming that one-fifth of the breaking load of the chain 
comes upon the arms. Equating the bending moment per arm 
to its resisting moment, considering the arm to be extended to the 



Art. 180] 



BLOCK CHAINS 



239 



center of the shaft, we have, assuming the arm to have an ellipti- 
cal cross-section, 

WJ) = irSbh 2 

10 n 32 ' 
from which 

WD 

— ~- (approximately) (277) 



bh- 



The arms of sprockets are generally made with a cross-section 
approximating an ellipse having a ratio between the major and 
minor axis of about 2.5 to 1 at the center of the sprocket. At the 
rim, the major and minor axes are made 0.8 and 0.3, respectively, 
of the major axis at the center. 

An investigation of actual sprockets based upon the above 
assumptions showed that S varied from 2,500 to 3,300 pounds 
per square inch, in round numbers. As an average value use 
3,000. Letting b = 0.4 h, (277) becomes 



<f 



2.5 WD 



nS 



(278) 



POWER CHAINS 

The types of chains discussed in the preceding articles of this 
chapter are not well adapted to any service requiring speeds 



Fig. 103. 



P 



above 600 feet per minute, and for that reason they are not suit- 
able for the transmission of power where the speed exceeds this 
limit. For this class of service special forms of chains, all parts 
of which are machined fairly accurately, have been devised. 
These may be classified as follows: (a) block chains; (b) roller 
chains; (c) silent chains. 

180. Block Chains. — As the name implies, the block chain, 
shown in Fig. 103, consists of solid steel blocks shaped like the 



240 



TABLE OF DIAMOND BLOCK CHAINS [Chap.X 



letter B or the figure 8, to which the side links are fastened by- 
hardened steel rivets. Block chains have proven very satis- 
factory for light power transmission where the speeds do not ex- 
ceed 800 to 900 feet per minute. Table 58 gives the commercial 
sizes of the block chains manufactured by the Diamond Chain 
and Mfg. Co. 

Table 58. — Diamond Block Chains 





Pitch 


Dimensions 


Width' 

of 
block 


Diam. 

of 
rivet 


Weight 
per 
foot 


Ultimate 
strength 


Chain No. 


a 


6 


h 


102 


1 


0.400 


0.600 


0.325 


Vie 

% 




0.33 
0.38 
0.42 
0.50 


1,500 
1,600 
1,800 
2,000 


103 


1 


0.400 


0.600 


0.325 


Vie 

% 

V2 




0.33 

0.38 
0.42 
0.50 


2,200 
2,300 
2,400 
2,500 


105 


IK 


0.564 


0.936 


0.532 


% 


0.265 


0.89 
1.03 


5,000 



181. Sprockets for Block Chains. — In Fig. 104 is shown a 
design of a block chain sprocket, the rim part being made of steel 
plate bolted on to a cast-iron hub. Instead of using the built- 
up construction, the sprocket may be made completely of cast 
iron with a central web, or with arms, if the sprocket is large in 
diameter. Denoting the pitch diameter of the sprocket by the 
symbol D, the number of teeth in the sprocket by T, and the 
pitch of the chain by p, then the magnitude of the angle a shown 
in Fig. 104 is given by the following expression: 



180 c 
T 



(279) 



From the geometry of the figure, it follows that 

a 



sin (a — j3) 



and 



D 

*- 

D 



(280) 
(281) 



Art. 181] 



BLOCK CHAIN SPROCKETS 



241 



Deriving an expression for /3 by eliminating D, we have 

sin a 



tan 



(282) 



cos a + 



To obtain the pitch diameter of the sprocket for any desired 
number of teeth and given size of chain, determine the angle a 
and substitute this angle in (282) in order to establish the angle 
/3. Knowing /3, the pitch diameter D may be found by means of 
(281). To get satisfactory service from sprockets, the minimum 



r-t> 




Fig. 104. 

number of teeth should be limited to 15, unless the rotative 
speed of the sprocket is low. The teeth of small sprockets have 
a tendency to wear hook-shaped, thus causing noise and at the 
same time decreasing the life of the installation. 

The height of the tooth is usually made slightly greater than 
the dimension h in Table 58. It should be noted that the space 
between the teeth is made somewhat longer than the overall 
length of the block, in order to provide for the stretching of the 
chain due to wear on the rivets. 



max. p 

1 \ N I 



(283) 



242 SELECTION OF BLOCK CHAINS [Chap. X 

182. Selection of Block Chains. — A careful study of the opera- 
tion of chains of the block and roller type conducted by the 
Diamond Chain and Mfg. Co. indicates that the noisy operation 
and the rapid wear of a chain are due chiefly to the impact be- 
tween the sprocket and the rollers or blocks as the latter seat 
themselves. The effect of impact is more marked when a long 
pitch chain runs over a sprocket having a high rotative speed. 
As a result of this study, the following empirical formulas and 
rules have been proposed by the Diamond Chain and Mfg. Co. : 

/900y 

\N I 

max. N of small sprocket = —= 

Vp 3 

(a) In an installation in which the load on the chain is fairly 
uniform, the permissible chain pull should not exceed one-tenth 
of the ultimate strength of the chain as given in Table 58. 

(6) As a further check on the chain load, the equivalent pres- 
sure per square inch of projected rivet area should not exceed 
1,000 pounds for general service. When slow chain speeds pre- 
vail, this pressure may run as high as 3,000 pounds, although the 
latter value should be considered the upper limit. 

(c) When the chain is subjected to sudden fluctuations of load, 
the permissible chain pull may only be ^o or 3^o °f * ne ultimate 
strength. 

(d) In selecting a block or roller chain for a given duty it is well 
to give preference to a light chain rather than a heavy one, pro- 
vided the former has sufficient rivet area as well as strength to 
transmit the power. As stated above, long life and quiet run- 
ning are secured more easily by selecting a short pitch chain. 
As a rule, a narrow chain is more satisfactory than a wide one 
except in places where the sprockets are not always in proper 
alignment; for example, in an electric motor drive or in motor- 
truck service. 

183. Roller Chains. — A typical roller chain is shown in Fig. 
105. This type of chain is used to some extent in motor-vehicle 
service, especially on trucks, as well as for general power trans- 
mission. Chain speeds as high as 1,400 feet per minute have 
been used successfully on light loads; but for general use with 
proper lubrication 1,200 feet per minute should be the limit. 
Occasionally double roller chains are used and if properly in- 



Art. 184] 



ROLLER CHAIN SPROCKETS 



243 



stalled they give good service. In Table 59 are given the com- 
mercial sizes and other information pertaining to the roller 
chain made by the Diamond Chain and Mfg. Co. Instead of 
the ultimate strength of the chain, the normal and maximum 
allowable loads are given. The normal loads are based on a 
bearing pressure of 1,000 pounds per square inch of the projected 
area of the rivet, while the maximum load is approximately three 
times the normal but in no case will it exceed one-tenth of the 
ultimate strength of the chain. 

In arriving at the size of a roller chain required for a particular 
duty, the various points mentioned in Art. 182 apply equally 
well in the present case. 




184. Sprockets for Roller Chains. — As in the case of block 
chains, the sprockets used with high-grade roller chains are always 
made with cut teeth. The forms given to the teeth by the vari- 
ous manufacturers of roller chains differ considerably. 

(a) Old-style tooth form. — In Fig. 106 is shown a tooth form 
that is faulty in that it makes no provision for the stretching of 
the chain due to wear on the pins or rivets. If the space between 
the teeth were made wider, as shown in Fig. 108(a), giving the 
roller more clearance, the chain drive would be satisfactory. At 
the present time cutters that give a clearance approximating one- 
tenth of the radius of the chain roller are used in the manufacture 
of sprockets. As the chain runs on or off the sprocket, the curve 
described by the roller is an involute of the pitch circle, from 
which it would appear that the face of the tooth should be made 
an involute. This, however, is not done as the face of the tooth 
is generally made an arc of a circle a trifle inside of the involute 



244 



TABLE OF DIAMOND ROLLER CHAINS [Chap. X 



in order that the roller will have no contact with the tooth on 
entering or leaving the sprocket. The length of the addendum 
of the tooth is arbitrarily taken as one-half of the diameter of 
the roller. The pitch diameter of the sprocket is obtained by 
the use of formula (275) derived for the common detachable 
chain in Art. 178. 







' Table 59 — 


-Diamond Roller Chains 




Chain 
No. 


Pitch 


Roller 




Weight 
per foot 


Allowable load 




Length 


Diam. 


rivet 


Normal 


Maximum 


Remarks 


75 


M 


■h 

He 
H 


0.306 


l H* 


0.280 
0.300 
0.320 


44 
55 
65 


120 


Single roller 


147-149 


% 


y* 


0.4 


0.200 


0.475 
0.619 


83 
108 


250 
325 


Single roller 


151 


m 


y% 

3£ 


/■ '8 


0.312 


1.580 
1.690 
1.800 


253 
292 
331 


760 

877 
994 


Single roller 


153 


H 


He 

% 

y 
y* 


0.469 


0.220 


0.710 
0.760 
0.860 
0.960 


106 

120 
147 
175 


317 
359 
442 
500 


Single roller 




1.450 


295 


750 


Double roller 


154 


l 


H 

% 
H 
H 


/■£ 


0.312 


1.680 
1.810 
1.940 


253 

292 
331 


760 
877 
994 


Single roller 




3.290 


585 


1,700 


Double roller 


155 


l 


H 

y 

y 


%6 


0.281 


1.070 
1.170 
1.270 


176 
211 
246 


527 

632 
738 


Single roller 




1.840 


422 


1,200 


Double roller 


157 


1M 


i 


H 


0.375 


2.410 
2.740 


396 
492 


1,189 
1,476 


Single roller 


160 


m 


H 

l 




0.375 


2.540 
2.690 
2.990 


350 
396 
492 


1,049 
1,189 
1,476 


Single roller 




4.850 


793 


2,400 


Double roller 


162 


m 


H 

l 


% 


0.437 


3.890 
4.150 


520 
629 


1,560 

1,888 


Single roller 


164 


m 


l 


l 


0.500 


4.960 


720 


2,160 


Single roller 


8.750 


1,440 


4,000 


Double roller 


168 


2 


1>4 


iy 


0.5625 


6.320 


975 


2,925 


Single roller 


11.560 


2,231 


6,000 


Double roller 



(6) Diamond tooth form. — In Fig. 107 is shown the method used 
by the Diamond Chain and Mfg. Co. for laying out their latest 
type of sprocket. The information given in the figure as well as 
the formula below were kindly furnished by Mr. G. M. Bartlett, 



Art. 184] 



ROLLER CHAIN SPROCKETS 



245 



mechanical engineer for the firm. In the following formulas p 
represents the pitch of the chain as shown in Fig. 105. 





Fig. 106. 

a = chain width — 0.045 p 

b = 0.545 p 

c = 0.3 p 

d = diameter of roller 



(284) 



The angle of pressure between the roller and the tooth is 20 
degrees, as shown in the figure. 




(c) Renold tooth form. — Another recent design of sprocket tooth 
form is illustrated in Fig. 108(&). It represents the results of 
many years of experience with roller chains as well as several 
years of special research work by Mr. Hans Renold, a prominent 



246 



LENGTH OF ROLLER CHAIN 



[Chap. X 



English chain manufacturer. The results of his work were pre- 
sented before the American roller chain manufacturers in the 
spring of 1914. The form of the tooth, which is not protected 
by patents, has a distinct advantage over the older forms still 
used by some chain makers, in that the stretch of the chain is 
taken care of by the rollers rising on the tooth flanks. The tooth 
is thus prevented from wearing into a hook form and a smooth- 
running transmission is insured. 

The space between the teeth is made an arc of a circle having a 
radius equal to the diameter of the roller or a few thousands of an 





Fig. 108. 



inch larger. The straight lines forming the teeth are tangent to 
this arc and make an angle of 60 degrees with each other as shown 
in the figure. The face of the tooth is relieved near the top by a 
circular arc. The height of the tooth thus formed is greater than 
that used with other tooth designs. 

185. Length of Roller Chain. — It is evident that a chain cannot 
have a fractional number of pitches or links ; hence in all cases the 
next whole number above the calculated number must be selected, 
and if the distance between the centers of the driving and 
driven sprocket will permit a slight change, the number chosen 
should be an even number. An odd number of pitches will 
necessitate the use of an offset link for joining the ends of the 
chain. The following formula used by the Diamond Chain and 



Art. 186] SILENT CHAINS 247 

Mfg. Co. gives the chain length in pitches and has been found to 
give accurate results : 

Chain length) =2L j r y {Tl + r ) + 0.0257 _ } 

in inches j ' L 

in which L denotes the distance between the centers of the two 
sprockets, and Ti and T 2 the number of teeth on the large and 
small sprocket, respectively. If it is desirable to determine the 
length of the chain in inches, merely multiply the pitch by the 
chain length obtained from (285). 

186. Silent Chains.' — The best forms of chain capable of 
transmitting power at high speeds are those designated as 
silent chain. An installation of such a chain if properly designed 
and constructed will be just as efficient as a gear drive for the 
same conditions of operation. At the present time there are in 
use several designs of silent chain, having in general the same 
form of link and differing only in the type of joint used. With 
silent chains, the load transmitted is distributed equally between 
all of the sprocket teeth in contact with the chain, and is not 
carried by a single tooth as is the case in some of the chains here- 
tofore discussed. 

Silent chains are well adapted for transmitting power economic- 
ally at speeds of 1,200 to 1,500 feet per minute. The lower speed 
holds for chains having a pitch greater than one inch and the 
higher value for small chains. If the speed is in excess of 1,500 
feet per minute, chains are liable to be noisy unless they are 
enclosed and run in oil. With properly designed gear cases 
and with the use of good lubricants, the smaller sizes of chains 
may be run at 2,000 feet per minute and the larger sizes at 1,500. 
It should be borne in mind, however, that these speeds are 
attained at the cost of reduced life of the chain. Where a positive 
drive is essential, as in direct-connected motor-driven machinery, 
and where the shafts are too far apart for gearing, silent chains 
are used extensively. Chains transmitting power in dusty and 
dirty surroundings should always be enclosed in an oil-tight case. 

187. Coventry Chain. — The Coventry chain shown in Fig. 109 
is manufactured in England, but is used to a considerable extent 
in America. It consists of links of special form assembled in 
pairs and held together by the hardened steel bushes b. Various 
widths of chains are produced by assembling these double links 



248 



WHITNEY CHAIN 



[Chap. X 



alternately on hardened steel pins; for example, the chain shown 
in Fig. 109 is called a 1 X 2 combination. The links themselves 
are not hard, and their shape is such that the load is distributed 
equally over all the teeth on the sprocket in actual contact with 




the chain. This action is illustrated in Fig. 110, which also shows 
the form of tooth used on such sprockets. 

188. Whitney Chain. — The chain illustrated by Fig. Ill is an 
American design, manufactured by The Whitney Mfg. Co. of 




Fig. 110. 



Hartford, Conn. The shape of the links in this chain is similar to 
that used on the Coventry chain, and hence the action of the links 
on the sprocket teeth is practically the same. The individual 
load links turn on the outside of the hard steel bushes b which are 
fastened securely into the guide plates a. The hardened steel 



Art. 189] 



LINK BELT CHAIN 



249 



pins turning within the bushings are forced into outside steel 
plates shaped like the figure eight. The function of the outside 
plates is to increase the tensile strength of the chain. 




Fig. 111. 

189. Link Belt Chain.— The Link Belt Co., after manufactur- 
ing for several years a plain pin-joint silent chain patented by 
Hans Renold of England, finally introduced the chain illustrated 
in Fig. 112. The joint consists of a case-hardened steel pin hav- 
ing a bearing on two case-hardened steel bushes b and c. These 




bushes are segmental in shape and are fitted into broached holes 
in the links, as shown in the figure. This type of joint increases 
the bearing area on the pin over that obtained in the original 
Renold chain that had no bushes at all. This chain is not pro- 
vided with guide plates, so special provisions must be made on 
the sprocket for retaining it. 



250 



MORSE CHAIN 



[Chap. X 



190. Morse Chain. — In the Morse chain shown in Fig. 113, the 
joint is of a peculiar construction in that it introduces rolling fric- 
tion in place of the sliding friction common to all the types of 
silent chains discussed in the preceding articles. The joint con- 
sists of two hardened steel pins b and c anchored securely in their 
respective ends of the link. The pin b has a plane surface against 
which the edge of the pin c rolls as the chain runs on or off the 
sprocket. The wear all comes upon the two pins and these may 
be easily renewed. When the chain is off the sprocket the load 
upon the joints for that part of the chain between the sprockets 




Fig. 113. 



is taken by relatively flat surfaces and not by the edge of the pin 
c. It is probable that the Morse chain will give better service in 
dusty places than any other type of silent chain, due to the fact 
that the rocker joint used requires less lubrication than the cylin- 
drical pin joints. 

191. Strength of Silent Chains. — The life of a silent chain de- 
pends upon the bearing area of the pins or bushings and not so 
much upon the ultimate strength. For minimum wear of the 
chain and for maintained efficiency, the working load under 
normal conditions approximates one-thirtieth of the ultimate 
strength, while under severe fluctuations of load at the maximum 
speed it is taken as one-fiftieth of the ultimate strength. Some 
manufacturers limit the bearing pressure on the pins to 650 
pounds per square inch of projected pin area. Since the strength 



Art. 192] SILENT CHAIN SPROCKETS 251 

of a chain can be increased by merely adding to its width, it is 
evident that for the same load conditions, chains of different 
pitches and widths may be selected; for example, a 1-inch pitch 
chain 4 inches wide and a 1^-inch pitch chain 3 inches wide are 
capable of transmitting approximately the same horse power at 
the same speed. Experience dictates that the width should range 
from two to six times the pitch. 

The first cost of narrow chains having a long pitch is less than 
wide ones of a shorter pitch. The longer pitch chains require 
larger sprockets, but are to be preferred when the distance between 
the connected shafts is great. Frequently it is found desirable 
to run two chains side by side in order to transmit the desired 
horse power. 

In Table 60 is given information pertaining to the Morse 
chain, which will serve for making the preliminary study 
of a silent-chain installation. This information was kindly fur- 
nished by the Morse Chain Co. of Ithaca, N. Y. Table 61 con- 
tains useful data relating to the Whitney chain, while Table 62 
applies to the Link Belt chain. 

192. Sprockets for Silent Chains.— An inspection of the 
figures illustrating the various types of silent chains shows that 
the shapes of the individual links are all about alike. The angle 
included between the working faces of the link is made 60 degrees 
by all of the manufacturers; hence it follows that the angle in- 
cluded between the flanks of alternate teeth will always be 60 
degrees irrespective of the number of teeth in the sprocket. 
However, the angle included by the flanks of the same tooth will 
change, being small for lower numbers of teeth. As this angle 
decreases rapidly for sprockets having small numbers of teeth, 
the manufacturers try to limit the number of teeth in small 
sprockets to 15. Whenever the installation permits, and when 
very quiet operation is desirable, the lower limit is placed at 17. 
Again, since the angle between the flanks of the same tooth in- 
creases with the number of the teeth in the sprocket, it is found 
necessary to limit the number of teeth to about 120 or 130 on 
account of the liability of the chain to slide over the teeth. The 
tooth form for any particular make of chain is determined best 
by laying it out to conform to the dimensions of the links to be 
used. 

The so-called pitch diameter of the sprocket for silent chain 



252 



MORSE. CHAIN DATA 



[Chap. X 



£ rtio ooo m 

M07HH10010100 ©■* 






OOO 00 

Nh",'01HKOOiO«1 OlO 
.-ICO J 1M I O^MI^COOOOO 



NtJHOCM 



tj 1-1 1» 000 m 

■ 9VOHNOO(-H ©CO 

io^n 1 TtintoMcqioooo 

1-1 lOO© HCCOH 



JjJ.lNjjCOINQOINT-ICCunO^ 



52 gco 000 co 



H""0 Q ' H 



OHOO 



S ^O ©CO OS 

h <N (J. rH Jj CM .h NH NUJOra 
■- 1 l «OO ri OHOO 



^OlOOOO COO 

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10 CM 



H "OO^ 



- 1 rt 10 « in -- 1 '- 1 - 



SSVgT^ 



qS 



JMOOO 
JWNOH 

dodo 



Ml -W "" S 



.3 c-S_fl2' Q 



22 . 

ftflfi 



•ff. 

'3*0 ° 

■5 o ft ^ 

--Sag 1 

T*.a -I 



mill. 1 «g 1 

-3 .fa 3. fa « 3 m a M ™ 

r^ a)" OJ A l5 E 03tT o 



ftp, 



ftp, 
5 ° 



CM lO 



II II II 



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'3*3 
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HHiaHWMiaHSlHWHH 
H H H H H-S H E* H g H H H 

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Art. 192] 



TABLE OF WHITNEY CHAINS 



253 



having pin joints may be determined by the formula used for 
roller chains, namely 

V , . 
sin a 



D 



in which p denotes the pitch of the chain and the angle a is 
equal to 180 degrees divided by the number of teeth. Whenever 
possible an odd number of teeth should be used for the pinion so 
that the wear may be distributed more evenly. Sprockets should 
be made as large as possible to relieve the wear on the chain, as 
in passing around small sprockets the angular displacements of 
each link on the pins or bushes is greater than in the case of large 
sprockets. 









Table 61 — 


-Whitney Silent Chains 






6 
'3 

8 


A 

£ 


$2l 


a 


1.1 

lis 


If 

+3 g 

5-s 


6 
a 
6 


s 


c.2 


ft 
'3 


ft 
J I" 

si's 




1201 




X 


0.56 




2,800 


1265 




m 


3.22 




14,400 


1202 




% 


0.74 




3,400 


1266 




2 


3.59 




15,600 


1203 


% 


l 


0.92 




4,000 


1267 


2 ' ,, 


3.96 




16,800 


1204 




m 


1.10 




4,600 


1268 




2H 


4.33 




18,000 


1205 




ix 


1.28 




5,200 


1269 
1270 
1271 




2% 

3 

3X 


4.70 
5.07 
5.44 




19,200 
20,400 
21,600 


1221 




x 


0.83 




4,900 


u 




1222 




H 


1.08 




5,800 


1272 




3M 


5.81 




22,800 


1223 




i 


1.33 




6,700 


1273 




3% 


6.18 




24,000 


1224 


x 


IX 


1.58 




7,600 


1274 




4 


6.55 




25,200 


1225 




IX 


1.83 




8,500 


1275 




4K 


6.92 




26,400 


1226 




1% 


2.08 




9,400 


1276 




4^ 


7.29 




27,600 


1227 




2 


2.33 




10,300 


















1281 
1282 




1 

IX 


3. .24 
4.25 




17,200 
20,100 


1241 




H 


1.35 




7,100 






1242 




l 


1.66 




8,000 


1283 




2 


5.26 




23,000 


1243 




IX 


1.97 




8,900 


1284 




2^ 


6.27 




25,900 


1244 




IX 


2.28 




9,800 


1285 




3 


7.28 




28,800 


1245 




\H 


2.59 




10,700 


1286 




3X 


8.29 




31,700 


1246 




2 


2.90 




11,600 


1287 




4 


9.30 




34,600 


1247 


*A 


2K 


3.21 




12,500 


1288 


1 


4M 


10.31 




37,500 


1248 




2M 


3.52 




13,400 


1289 




5 


11.32 




40,400 


1249 




2% 


3.83 




14,300 


1290 




5X 


12.33 




43,300 


1250 




3 


4.14 




15,200 


1291 




6 


13.34 




46,200 


1251 




3K 


4.45 




16,100 


1292 




6J^ 


14.35 




49,100 


1252 




3X 


4.76 




17,000 


1293 
1294 
1295 




7 

7M 

8 


15.36 
16.37 
17.38 




52,000 
54,900 
57,800 


1261 




H 


1.74 




9,600 






1262 




l 


2.11 




10,800 














1263 


H 


IN 


2.48 




12,000 














1264 




IX 


2.85 




13,200 















254 



TABLE OF LINK BELT CHAINS 



[Chap. X 



Table 62. — Horse Power Transmitted By Link Belt Silent Chain 


*°.S 


o d 








Speed of ch 


ain in ft. per min. 






500 


600 


700 


800 


900 


1,000 


1,100 


1,200 


1,300 


1,400 


1,500 




K 


0.58 


0.66 


0.72 


0.78 


0.82 


0.88 


0.91 


0.95 










% 


0.87 


0.98 


1.07 


1.16 


1.22 


1.30 


1.38 


1.42 










1 


1.16 


1.31 


1.43 


1.55 


1.63 


1.73 


1.82 


1.89 








% 


IK 


1.45 


1.64 


1.79 


1.91 


2.04 


2.18 


2.28 


2.36 










IK 


1.74 


1.97 


2.15 


2.30 


2.45 


2.60 


2.73 


2.83 










2 


2.32 


2.62 


2.86 


3.08 


3.27 


3.46 


3.64 


3.78 










3 


3.48 


3.91 


4.28 


4.61 


4.89 


5.22 


5.46 


5.67 










K 


0.84 


0.95 


1.04 


1.11 


1.19 


1.27 


1.33 


1.38 


1.42 








H 


1.26 


1.40 


1.56 


1.70 


1.79 


1.91 


1.99 


2.07 


2.13 








l 


1.68 


1.89 


2.08 


2.25 


2.34 


2.54 


2.65 


2.76 


2.84 






K 


IK 


2.52 


2.91 


3.12 


3.44 


3.57 


3.88 


3.98 


4.14 


4.25 








2 


3.37 


3. 82 


4.17 


4.48 


4.77 


5.10 


5.30 


5.52 


5.68 








3 


5.05 


5.73 


6.25 


6.75 


7.15 


7.60 


7.95 


8.29 


8.50 








4 


6.73 


7.64 


8.30 


9.00 


9.53 


10.10 


10.60 


11.10 


11.30 








1 


2.22 


2.51 


2.74 


2.96 


3.15 


3.33 


3.50 


3.64 


3.75 








IK 


2.77 


3.15 


3.41 


3.71 


3.93 


4.18 


4.37 


4.54 


4.70 








m 


3.33 


3.76 


4.12 


4.43 


4.72 


5.00 


5.25 


5.45 


5.62 






% 


2 


4.43 


5.02 


5.47 


5.91 


6.30 


6.67 


7.00 


7.28 


7.50 








3 


6.65 


7.52 


8.22 


8.88 


9.45 


10.00 


10.50 


10.90 


11.20 








4 


8.86 


10.00 


10.90 


11.80 


12.60 


13.30 


14.00 


14.50 


15.00 








6 


13.30 


15.00 


16.40 


17.70 


18.90 


20.00 


21.00 


21.80 


22.50 








I 


2.85 


3.22 


3.51 


3.78 


4.05 


4.37 


4.48 


4.65 


4.82 








IK 


3.56 


3.98 


4.39 


4.70 


5.06 


5.30 


5.60 


5.78 


6.02 








IK 


4.27 


4.85 


5.27 


5.67 


6.10 


6.40 


6.72 


6.98 


7.23 








2 


5.68 


6.42 


7.03 


7.56 


8.10 


8.55 


8.95 


9.31 


9.63 






% 


3 


8.55 


9.63 


10.50 


11.40 


12.10 


12.80 


13.40 


14.00 


14.50 








4 


11.40 


12.80 


14.00 


15.10 


16.30 


17.30 


17.90 


18.60 


19.30 








5 


14.20 


16.10 


17.60 


18.90 


20.30 


21.30 


22.40 


23.30 


24.10 








6 


17.10 


19.30 


21.10 


22.80 


24.30 


25.70 


26.80 


27.90 


28.90 








2 


7.00 


7.91 


8.65 


8.33 


10.00 


10.50 


10.90 


11.40 


11.80 








2K 


9.00 


10.10 


11.10 


12.00 


12.90 


13.50 


14.10 


14.70 


15.20 








3 


11.00 


12.40 


13.60 


14.60 


15.70 


16.50 


17.20 


18.00 


18.60 






1 


4 


15.00 


16.90 


18.60 


20.00 


21.50 


22.50 


23.50 


24.60 


25.40 








5 


19.00 


21.50 


23.50 


25.20 


27.20 


28.70 


29.70 


31.10 


32.10 








6 


23.00 


26.00 


28.50 


30.50 


32.90 


34.50 


36.00 


37.60 


38.90 








8 


31.00 


34.90 


38.40 


41.20 


44.30 


46.30 


48.50 


50.70 


52.40 








2 


9.70 


11.00 


11.90 


13.00 


13.80 


14.60 


15.30 


15.90 


16.40 


16.7 






3 


15.30 


17.30 


18.70 


20.30 


21.70 


22.90 


24.20 


25.00 


25.70 


26.5 






4 


20.80 


23.50 


25.50 


27.60 


29.60 


31.20 


32.60 


34.10 


35.10 


36.2 




1H 


5 


26.30 


29.80 


32.30 


35.10 


37.50 


39.70 


41.60 


43.20 


44.50 


45.8 






6 


31.80 


36.20 


39.10 


42.70 


45.30 


48.20 


50.30 


52.20 


53.80 


55.5 






8 


42.80 


48.50 


52.70 


57.20 


61.20 


64.00 


67.80 


70.30 


72.50 


74.6 






10 


54.10 


61.30 


66.50 


72.20 


77.10 


81.20 


85.60 


88.70 


91.40 


94.1 





Art. 192] 



TABLE OF LINK BELT CHAINS 



255 



Table 62. — Horse Power Transmitted by Link Belt Silent Chain (fiont.) 



"S 


"o 


Speed of chain in ft. per min. 


"o'S 






500 


600 


700 


800 


900 


1,000 


1,100 


1,200 


1,300 


1,400 


1,500 




3 


20.10 


22.70 


24.70 


26.90 


28.70 


30.30 


31.80 


33.00 34.00 


35. 


35.7 




4 


27.50 


31.10 


33.70 


36.60 


39.10 


41.20 


43.40 


45.00 


46.40 


48.0 


48.7 




5 


34.80 


39.30 


42.70 


46.30 


49.50 


52.30 


55.00 


57.00 


58.70 


60.7 


61.6 


1« 


6 


42.20 


47.60 


51.80 


56.30 


60.00 


63.40 


66.50 


69.00 


71.10 


73.5 


74.7 




8 


56.70 


64.20 


69.70 


75.70 


81.00 


85.20 


89.70 


93.00 


95.80 


99.0 


101.0 




10 


71.40 


80.70 


87.70 


95.20 


102.00 


107.00 


113.00 


117.00 


121.00 


124.0 


127.0 




12 


86.00 


97.30 


106.00 


115.00 


123.00 


129.00 


136.00 


141.00 


145.00 


150.0 


152.0 




6 


56.10 


63.50 


69.00 


75.00 


80.00 


84.30 


88.80 


92.00 


94.80 


97.5 


99.6 




8 


75.70 


85.60 


93.00 


101.00 


108.00 


114.00 


120.00 


124.00 


128.00 


131.0 


134.0 


2 


10 


95.20 


107.00 


117.00 


126.00 


136.00 


143.00 


151.00 


156.00 


161.00 


165.0 


169.0 




12 


114.00 


129.00 


141.00 


153.00 


164.00 


172.00 


182.00 


188.00 


194.00 


199.0 


204.0 




14 


134.00 


152.00 


165.00 


179.00 


191.00 


201.00 


212.00 


220.00 


227.00 


233.0 


240.0 




16 


154.00 


174.00 


189.00 


205.00 


220.00 


231.00 


243 . 00 


252.00 


260.00 


267.0 


273.0 




6 


73.00 


82.70 


90.00 


98.00 


104.00 


110.00 


116.00 


120.00 


124.00 


127.0 


130.0 




8 


100.00 


113.00 


123.00 


133.00 


143.00 


150.00 


158.00 


164.00 


169.00 


174.0 


178.0 


2M 


10 


126.00 


143.00 


155.00 


168.00 


180.00 


190.00 


200.00 


207.00 


213.00 


220.0 


224.0 




12 


153.00 


173.00 


188.00 


204.00 


218.00 


230.00 


242.00 251.00 


259.00 


266.0 


272.0 




14 


179.00 


204.00 


220.00 


240.00 


255.00 


270.00 


284.00 294.00 


303.00 


313.0 


318.0 




16 


206.00 


235.00 


253.00 


274.00 


294.00 


310.00 


326.00 338.00 


348.00 


359.0 


365.0 



The several makes of so-called silent chains require different 
types of sprockets in order to keep the chain from running off, as 




Fig. 114. 



may be noticed by consulting Figs. 114 and 115. The so-called 
outside-guided chain shown in Figs. 109 and 111 require 
plain sprockets, since the guide links prevent it from running off. 
The Link Belt chain, having no guide links, depends upon flanged 



256 



SPRING-CUSHIONED SPROCKETS 



[Chap. X 



sprockets of one form or another. One design of such a sprocket, 
as used by the Link Belt Co., is shown in Fig. 115, and in Table 63 
are given some general proportions pertaining thereto. The 
Morse chain is always provided with central guide links; hence, 
the sprocket teeth are provided with one or more central grooves 
in which the guide plates run. A design of this description is 
shown in Fig. 117. 

c - 




Fig. 115. 
Table 63. — General Proportions of Link Belt Sprockets 









Dimensions 






Chain 












pitch 














a 


b 


c 


e 


/ 


% 


He 


y% 




%2 


iMi 


y* 


0.2 


Vie 




H 


all 


% 


0.25 


x Yxe 






H 


0.3 


% 


2e +f 


%2 


1 o| 


l 


0.4 


V4 






M 


0.5 


1% 




%2 


|S| 


1H 


0.6 


1% 






^ s g 


2 


0.85 


2He 


>. 


He 


•s 2 £ 


2V 2 


1.25 


2V 2 




He 





193. Spring-cushioned Sprockets. — In a power transmission 
subjected to shocks due to intermittent and irregular loads, it is 



Art. 193] 



SPRING-CUSHIONED SPROCKETS 



257 



considered good practice to use a form of sprocket that is capable, 
of absorbing these shocks thereby relieving the chain. In general, 
such a device (see Fig. 116 or 117) consists of an inner hub a 
keyed to the shaft, and upon this hub is mounted the sprocket 
rim e. Between the lugs b, cast integral with a, and the lugs 
d on the inside of the rim e are placed the compression springs c, 
through which the driving load must be transmitted. The design 
shown in Fig. 116 is furnished with a cover plate / to make it 
dustproof, and is representative of the practice of the Link Belt 




Fig. 116. 



Co. The Morse Chain Co. spring-cushion sprocket, shown in 
Fig. 117, is also dustproof but the split-rim construction is used. 

It is suggested that spring-cushioned sprockets are well adapted 
to such service as is met with in driving air compressors, pumps, 
metal planers and shapers, and punching and shearing machinery ; 
however, they are not used to any extent in such places, no doubt 
due to the additional cost. 

Whenever two chains are used side by side to transmit a given 
horse power, a "compensating sprocket" should be used unless the 
transmission is horizontal and the distance between the shafts is 
considerable so that quite a little weight of chain is between the 
sprockets. A compensating sprocket may be made by mounting 



258 



REFERENCES 



[Chap. X 



two spring-cushioned sprockets side by side on one central hub, 
thus dividing the load equally between the chains. 

In the design of cushioned sprockets for intermittent work, 
for example, driving reciprocating pumps not subjected to a 
water-hammer or excessive overloading, the compressive load on 
the springs should be based on a chain load two and one-half to 
three times the actual load. In installations where water-ham- 
mers on pumps, or other heavy additional loads, would come upon 
the springs, the latter should be designed for loads from four to 
five times the actual load on the chain. 




Fig. 117. 



References 

Elements of Machine Design, by W. C. Unwin. 

Machine Design, Construction and Drawing, by H. J. Spooner. 

Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 

Mechanical Engineers' Handbook, by L. S. Marks. 

The Strength of Chain Links, Bull. No. 18, Univ. of Illinois Experiment 
Station. 

A Silent Chain Gear, Trans. A. S. M. E., vol. 23, p. 373. 

Roller Chain Power Transmission and Construction of Sprockets, Mchy., 
vol. 11, p. 287. 

Chart for Chain Drives, Amer. Mach., vol. 37, p. 854. 

Calculations for Roller Chain Drives, Mchy., vol. 20, p. 567. 

The Manufacture of Chain, Mchy., vol. 21, pp. 719 and 817. 

Roller and Silent Chain, Trans. Soc. of Auto. Engrs., vol. 5, p. 390. 

Silent Chain Power Transmission, 
Steel Elect. Engrs., September, 1914. 

The Transmission of Power by Chains, 
Engrs., November, 1914. 

Link Belt Silent Chain, Data Book, No. 125, Link-Belt Co. 

Power Chains and Sprockets, Diamond Chain and Mfg. Co. 

Diamond Tooth Form for Roller Chain Sprockets, Diamond Chain and 
Mfg. Co. 



Paper before the Assoc, of Iron and 
Birmingham Assoc, of Mech. 



CHAPTER XI 
FRICTION GEARING 

Friction gearing is employed when the positiveness of relative 
motion is either unnecessary or not essential. The wheels de- 
pend for their driving value upon the coefficient of friction of the 
composition wheel against its iron mate, and their actual driving 
capacity becomes a function of the pressure with which they are 
held in contact. This pressure is limited by the ability of the 
composition surface to endure it without injury. The composi- 
tion wheel should never be used as the driven member of a pair of 
wheels, since, being of a softer material, its surface would be 
injured and eventually ruined by the occasional rotation of the 
iron wheel against it under pressure before starting it from rest, 
or after an excessive load has brought it to a standstill. Friction 
gearing may be used for transmitting power between shafts that 
are parallel or between those that intersect. 

194. Experimental Results. — Several years ago an extended 
series of experiments on friction gearing was made at the labora- 
tory of Purdue University, the results of which were reported by 
Prof. Goss in a paper before the American Society of Mechan- 
ical Engineers. These experiments were made upon compressed 
strawboard driving wheels approximately 6, 8, 12 and 16 inches 
in diameter in contact with a turned cast-iron follower 16 inches 
in diameter. The pressures per inch of face varied from 75 to 
more than 400 pounds, and the tangential velocity from 400 to 
2,800 feet per minute. The following are some of the conclusions 
derived from these tests : 

(a) Slippage increases gradually with the load up to 3 per cent., 
and when it exceeds this value it is liable to increase very suddenly 
to 100 per cent., or in other words, motion ceases. 

(6) The coefficient of friction varies with the slip, and becomes 
a maximum when the slip lies between 2 and 6 per cent. 

(c) The coefficient of friction seems to be constant for all pres- 
sures up to a limit lying between 150 and 200 pounds per inch of 
face, but decreases as the pressure increases. 

259 



260 



COEFFICIENTS OF FRICTION 



[Chap. XI 



(d) The coefficient of friction is not affected by variations in 
the tangential velocity between the limits 400 and 2,800 feet per 
minute. 

(e) The coefficient of friction for the 6-inch wheel was about 
10 per cent, less than for the others. 

(/) A coefficient of friction of 20 per cent, is readily obtained 
with wheels 8 inches in diameter and larger. 

In December, 1907, Prof. Goss presented before the American 
Society of Mechanical Engineers, a second paper on the subject of 
friction drives, in which he reported the results of another exten- 
sive series of tests. The values of the coefficient of friction and 
permissible working pressure per inch of face for the various 
materials experimented with are given in Table 64. Pressures 

Table 64. — Experimental Data Pertaining to Friction Gearing 



Material 


Coefficient of friction- 
working values 


Safe 
working 
pressure 


Cast 
iron 


Alumin- 
num 


Type 

metal 


Leather 


0.135 
0.150 
0.150 
0.210 
0.255 
0.309 
0.330 


0.216 

0.183 

0.273 
0.297 
0.318 


0.246 

0.165 

0.186 
0.183 
0.309 


150 


Wood 


150 


Tarred fiber 

Cork composition . . 

Straw fiber 

Leather fiber 

Sulphite fiber 


240 
50 
150 
240 
140 



exceeding 150 pounds per inch of face may be used providing 
the conditions under which the wheels are working are known 
definitely, or where experience has proven their use permissible. 
Several manufacturers now make wheels that allow the use of 
working pressures of 250 pounds or more. 



SPUR-FRICTION GEARING 

195. Plain Spur Frictions. — The simplest form of friction gear- 
ing consists of two plain cylindrical wheels held in contact with 
each other by properly constructed bearings. Such wheels, 
shown in Fig. 118, are known as spur frictions. To determine 
the least pressure that must be applied at the line of contact in 
order that the gears may transmit a given horse power, the follow- 
ing method may be used : 



Art. 196] 



SPUR-FRICTION GEARING 



261 



Let H = the horse power transmitted. 

V = the mean velocity of the gears in feet per minute. 

/ = face of the gears. 

p = permissible pressure per inch of face. 

fj, = coefficient of friction. 

Evidently, the total radial pressure between the two wheels at 
the line of contact is fp, and the tangential force due to this 
pressure is fxfp. Now this 
force must at least equal 
the tangential resistance or 

T . ?M00H (2g6) 

Therefore, the least pres- 
sure required between the 
two spur frictions, so that 
H horse power may be 
transmitted is 




fp 



33,000 H 



(287) 



196. Applications of Spur Frictions. — Plain spur-friction gear- 
ing is used for driving light power hoists, coal screens, gravel 
washers, and various forms of driers . Another useful and interest- 
ing application of spur frictions is found in friction-board drop 
hammers used in the production of all kinds of drop forgings. 
Two designs, differing somewhat in the method of driving the 
friction rolls, are shown in Figs. 119 and 120. The methods of 
operation and control of the hammer are similar in the two de- 
signs. In Fig. 119, the friction rolls b and c are keyed rigidly to 
their respective driving shafts d and e and may be brought into 
contact with the board a, at the lower end of which is fastened 
the ram. It will be noticed that the friction rolls are brought 
into contact with the board a by rotating the eccentric bearings in 
which the driving shafts are supported. The bearings are ro- 
tated slightly by the rod/, which in turn is tripped by the descend- 
ing hammer. The ram and the various operating accessories are 
not included in the figure. 

The function of the friction rolls b and c is to return the ram to 
its initial position after a blow has been struck. As soon as the 
ram returns to its initial position, it lifts the rod / by means of a 



262 



ANALYSIS OF A DROP HAMMER 



[Chap. XI 



suitable mechanism, and consequently the friction board which 
will again drop unless it is held by the pawls g and h. These 
pawls are controlled by the operator through a treadle. 

The design shown in Fig. 119 is that used by the Billings and 
Spencer Co., and differs from the other in that both shafts d and 
e are mounted on eccentric bearings, each shaft being driven by a 
belt and pulley. Fig. 120 shows the general details of the design 

used by the Toledo 
Machine and Tool 
Co. The driving pul- 
leys are keyed to the 
shaft e, which has 
mounted upon it the 
roll c and a spur gear 
m. The latter 
meshes with the gear 
n which is fastened 
to the roll 6, both 
being mounted with 
a running fit on the 
shaft d. The shaft d 
is supported on ec- 
centric bearings by 
means of which the 
two rolls are brought 
in contact with the 
board a. 

In some designs of 
drop hammers, the 
teeth on the gears m 
and n are made of the 
buttressed type, since they transmit power in only one direction 
and at the same time are subjected to a considerable shock. 

197. Analysis of a Drop Hammer. — The total lifting force T 
exerted on the friction board by the driving rolls must exceed the 
weight Q of the ram so that it is possible to accelerate the latter 
at the beginning of the hoisting period. 

Let ti = number of seconds required to accelerate the ram. 
U = number of seconds during which the ram moves 
upward at constant velocity. 




Fig. 119. 



Art. 197] 



ANALYSIS OF A DROP HAMMER 



263 



t 3 = number of seconds required to bring the ram to 

rest after releasing the rolls. 
v = maximum velocity of ram during hoisting period. 

The hoisting period is really made up of three separate periods, 
namely: (1) the period during which the ram is accelerated; (2) 
the constant-speed period; (3) the period immediately following 
the releasing of the driving rolls, during which the ram gives up 
its kinetic energy. 

Using the above notation and assuming uniformly accelerated 
motion, the distance h 
travelled by the ram in 
its upward travel is 
given by the following 
expression : 

Denoting the ratio of T 
to Q by the symbol c and 
disregarding the fric- 
tional resistances, it is 
evident that the acceler- 
ating force is 

Qv 

gh 



Q(c - 1) 



from which 



h = W^T) (289) 

Substituting (289) in 
(288), and simplifying, 




from which 



2glc — 1J 

h _ v r c "I 
v ~ 2oLc - 1J 



(290) 



The total number of seconds required for the hoisting period is 

( ' + fe + ( s = ^[^r] + " ( 291 > 



264 GROOVED SPUR FRICTIONS [Chap. XI 

The number of seconds required by the ram to fall through the 
distance h is 

U = J— (292) 



Hence the time required for a complete cycle may be readily 
determined. 

During the accelerating period, the work W\ expended by the 
friction rolls upon the board of the ram is 

W > = W^T) (293) 

and during this same period, the useful work done is 

Ov 2 

Hence the lost work is 

w ' - %[^] ^ 

The work W represents the loss due to slippage which will 
tend to produce excessive temperatures, thereby charring the 
board of the ram; hence its magnitude must be kept down by 
using a speed v that is not too high, and by making c relatively 
large. In actual hammers, c varies from 1.2 to 2. 

The total lifting force T is produced by the pressure of the rolls 
upon the board and is given by the relation 

T = 2/xP, (296) 

in which P denotes the normal pressure between each roll and 
the board and n the coefficient of friction, which may be assumed 
to vary from 0.25 to 0.35. 

198. Grooved Spur Frictions. — In the case of plain spur fric- 
tions, the pressures upon the shafts are excessive for large powers, 
thus causing a considerable loss of power due to the journal fric- 
tion. To decrease this loss of power by decreasing the pressure 
upon the shafts, a form of gearing known as grooved spur fric- 
tions is used. Fig. 121(a) shows how such gears are formed. It 
is desired to determine the relation between the horse power 
transmitted and the total radial pressure between the frictions. 

Let P = radial thrust upon one projection or groove. 

R = total reaction on each side of projection or groove. 



Art. 198] 



GROOVED SPUR FRICTIONS 



265 



T = tangential resistance on each projection or groove. 
n = number of projections or grooves in contact. 
2 a = angle of the grooves. 

In Fig. 121(6) are shown the various forces acting upon one of 
the projections. From the force 
triangle ABC it follows that 

P = 2R sin (a + <p), (297) 

in which <p denotes the angle of 
friction as shown in the figure. 

In any type of friction gearing 
the tangential resistance T is 
equivalent to the coefficient of fric- 
tion multiplied by the total normal 
pressure at the line of contact, hence 
for the case under discussion 



T = 



33,000 H 

nV 



2Rsin(p, 



(298) 



in which H and V have the same 
meaning as in Art. 195. 

Eliminating the factor 2R by 
combining (297) and (298), the 
least total pressure nP between the 
two grooved friction gears is given 
by the following expression: 

nP = 

33,000g (sina + ^ cosa) (2Q9) 



From an inspection of Fig. 121 
(a), it is evident that along the lines 
of contact between the two gears, 
the so-called pitch point is the only 
one at which the two gears have 
the same peripheral speed. At 
all other points there is a difference 

in speed between the gears, and hence there must be slippage, as 
a result of which excessive wear might be expected. In order 
to make this difference in speed small and at the same time de- 
crease the resultant wear, the projections must be made com- 




266 



BEVEL-FRICTION GEARING 



[Chap. XI 



paratively short. Furthermore, the normal pressure per inch of 
side of groove or projection should not, according to Bach, ex- 
ceed 3,200 pounds. When a considerable number of grooves 
are used, it is necessary that they be machined very accurately 
or excessive wear due to high contact pressure will result. The 
angle 2 a of the grooves varies from 30 to 40 degrees. 




BEVEL-FRICTION GEARING 

Bevel frictions are used when it is desired to transmit power 
by means of shafts that intersect. Such gears are shown in 

Fig. 122. Referring to Fig. 
\ ^rr- ^ i 122, the gear marked 2 is 

keyed rigidly to its shaft 
while the gear 1 is splined 
to its shaft. By means of 
a specially designed thrust 
bearing operated by a lever 
or other means, the bevel 
gear 1 is brought into con- 
tact with gear 2 and held 
F IG . 122. there under pressure. In 

designing a bevel-friction 
transmission, both the starting and running conditions should be 
investigated. 

199. Starting Condition. — In the following analysis it is 
assumed that the transmission is to be started under full load, a 
condition met with frequently in connection with hoisting 
machinery. At the instant of starting, due to the relative motion 
between the surfaces in contact, the reaction R instead of being 
normal is inclined away from the normal by the angle of friction 
<p, as shown in Fig. 122. 

As in the preceding cases, the tangential force that can be trans- 
mitted by the two gears is equal to the product of the total normal 
pressure and the coefficient of friction ; thus 

H 



T = ixR cos <p = 33,000 

From the geometry of the figure it is evident that 
Pi P 2 



R 



sin (a + <p) cos (a + <p) 



(300) 



(301) 



Abt. 200] BEVEL-FRICTION GEARING 267 

Combining (300) and (301), the expressions for the least axial 
thrusts that come upon the gears are as follows: 

D 33,000 H . . . . , Qno . 

Pi = — — ^ — (sin a + ju cos a) (302) 

D 33,000 H , x /ono^ 

P 2 = — TT (cos a — /x sin a) (303) 

200. Running Condition. — After the transmission gets up to 
speed, the relative motion between the gears along the line of con- 
tact ceases; hence the reaction between the two surfaces in con- 
tact is normal. Calling this reaction R', we have the relations 

T , ™ ™ — H 



nR' = 


33,000 


V 


P'x 

sin a 


P* 
COS a 



R' 

Combining these equations and solving for P[ and P 2 we obtain 
the following: 

P[ = 33,000* Bin « (304) 

D , 33,000 H cos a , Qn -, 

P 2 = - m7 ^ C305) 

The expressions just derived may be put in slightly different 
form by substituting for sin a and cos a their equivalents in terms 
of the diameters Di and D 2 of the gears. The resulting forms are 

33,000 H T — 1 



P' = 33,00^? [-7=4=1 (306) 

M 7 [VDl + D\\ 



33,000 



- [VDl + D\\ 



(307) 



Equations (306) and (307) give the least thrusts required along 
the shafts of the transmission in order to transmit the given horse 
power. 

CROWN-FRICTION GEARING 

Crown-friction gearing is used to transmit power by means of 
shafts that intersect and are at right angles to each other. A 
simple form of this type of transmission as applied to the driving 
of a light motor car is shown in Fig. 123. In slightly modified 



268 



CROWN-FRICTION GEARING 



[Chap. XI 



form this same mechanism has been applied to machine tools for 
varying feeds. Within recent years crown frictions have been- 
used successfully in automobile, motor-truck, and tractor trans- 
missions, as well as in the driving of screw power presses and 
sensitive drilling machines. 

The wheel c in Fig. 123 is generally faced with compressed 
paper, vulcanized fiber, leather, or other suitable friction material, 




[Fig. 123. 

and is slightly crowned in order to decrease the slipping action 
which takes place, due to the varying speeds of the points in con- 
tact. Now since wheel c is made of a softer material than that 
used on the disc 6, it should act as the driver so that its surface 
will not be worn fiat at spots by the rotation of the disc against 
it under pressure. However, this is not the usual method of 



Art. 201] CROWN-FRICTION GEARING 269 

mounting a crown-friction transmission. As now installed, the 
disc serves as the driving member and in practically all cases its 
face is plain cast iron. In the design just mentioned, the speed 
of the wheel c may be varied by simply moving c across the face of 
the disc, while the direction of rotation of the wheel may be re- 
versed by moving it clear across the center of the disc. 

201. Force Analysis. — To determine the forces acting upon the 
various members of a crown-friction transmission similar to that 
shown in Fig. 123, the following method may be used: 

The twisting moment on the driving shaft is 63,030 ^ hence 

the tangential forces acting upon the driven wheel for the two 
limiting speeds are as follows: 

., ., . . , m 126,060 tf] 

At the minimum speed, I \ = 



At the maximum speed, T 



126,060 H 
ND 2 



(308) 



in which D x and D 2 denotes the minimum and maximum diam- 
eters of the driving disc, respectively. 

The thrusts that must be applied to the disc for the two speeds 
are obtained by dividing the values of Ti and T 2 by the coefficient 
of friction n, giving 

D 126,060 H ) 
" i 



f iND 1 

p _ 126, 060 H 

^ 2 ~ ~tfmT 



(309) 



The forces actually available on the chain sprocket / for the two 
cases considered are as follows : 



L>3 



(310) 



in which X) 3 denotes the diameter of the sprocket /. The 
efficiency 77 may be taken as 60 per cent, at the low speeds and 80 
per cent, at the high speeds. 

To determine the magnitude of the force F required to shift 
the driven wheel c, multiply the thrust P exerted by the disc b 
upon the wheel c by the coefficient of friction n and add to this the 



270 



CROWN-FRICTION GEARING 



[Chap. XI 



force required to overcome the frictional resistance between 
the wheel c and its shaft e. Whence 

F = P( M + j u 1 ), (311) 

in which ju x denotes the coefficient of friction between the wheel 
c and its shaft e. 

202. Pressures on the Various Shaft Bearings. — (a) Driving 
shaft. — The various forces discussed in Art. 201 produce pres- 
sures upon the several bearings used in the transmission. The 
same type of crown friction drive shown in Fig. 123 is represented 




diagrammatically in Fig. 124. The journal A on the driving 
shaft a, due to the tangential force 7\ and the thrust P at the 
point of contact is subjected to the following pressures: 

PD 

1. A horizontal force due to P, having a magnitude of 

This result is obtained by taking moments about axis of the 
journal B. 

2. A vertical force, due to T h the magnitude of which is 
obtained as in the preceding case. This pressure acts in 
the same direction as the tangential force T x and its magni- 
tude is — T\. 

m 

By a similar analysis the following forces acting upon the jour- 
nal B are determined: 



Art. 203] FRICTION SPINDLE PRESS 271 

PD 

3. A horizontal force equal to - 

1 m 

T 

4. A vertical force equal to — (m + n). 

From the analysis of the forces acting upon the shaft a, it is 
evident that at the point B this shaft is subjected to a bending 
stress in addition to a torsional stress. There may also be a com- 
pressive stress due to the thrust P, but this can be avoided by a 
careful arrangement of the thrust bearing at that point. To de- 
termine the size of the shaft, use the principles discussed in the 
chapter on shafting. 

(b) Driven shaft. — The driven shaft e is subjected to a combined 
torsion and bending between the wheel c and the sprocket /. 
The wheel c is acted upon by the two forces P and T\, the former 
producing pure bending of the shaft and the latter combined 
torsion and bending. After having calculated the load on the 
sprocket /, the pressures upon the bearings C and E may be 
obtained by an analysis similar to that used above. 

203. Friction Spindle Press. — The so-called friction spindle 
press used to a large extent in Germany is an excellent application 
of crown-friction gearing. In this country, the Zeh and Hahne- 
mann Co. of Newark, N. J. are about the only manufacturers that 
have introduced friction gearing on presses for forging and stamp- 
ing operations. One of their designs is shown in Fig. 125. The 
friction wheel d is really a heavy flywheel fastened rigidly to the 
vertical screw e. The face of the flywheel is grooved, and into 
this groove is fitted a leather belt which serves as the friction 
medium. The driving shaft a is equipped with two plain cast- 
iron discs b and c, which may be brought into contact with the 
friction wheel d by moving the entire shaft endwise. It should 
be understood that the function of the friction drive is merely to 
accelerate the flywheel d, and the energy stored up during the 
accelerating period does the useful work. To accelerate the fly- 
wheel, the driving shaft is moved endwise against the action of 
the spring / until b is in contact with d, thus causing the screw to 
rotate. This rotation causes the screw and attached flywheel to 
move downward, increasing its rotative speed as well as that in a 
downward direction. It should be noted that the flywheel gen- 
erates a spiral on the face of the disc b. At the end of the working 
stroke of the screw a suitable tappit, located on the crosshead at 
the lower end of the screw, operates a linkage which disengages 



272 



FRICTION SPINDLE PRESS 



[Chap. XI 



b and d, thus permitting the spring /to force the disc c against the 
flywheel d causing it to return to the top of the stroke. 

This type of press is especially adapted for work in which a hard 
end blow is required. It is not suitable for work requiring a 
heavy pressure through a considerable part of the stroke, such as 
is required in the manufacture of shells, for example. 




204. Curve Described by the Flywheel. — In discussing the 
action of the friction spindle press, it is of interest to investigate 
the nature of the path or curve described by the flywheel on the 
face of the friction disc. It is apparent that the tangential 
velocity v t of the flywheel rim is proportional to the radius of the 
driving disc; hence at any point a distance r from the center of 
the driving shaft, the magnitude of this velocity is 



v t = cr 



(312) 



The velocity v s of the screw in a direction parallel to its center 
line also is proportional to the radius r; hence 



kr 



(313) 



Combining (312) and (313), it follows that the ratio of v s to 
v t is a constant, the value of which is readily determined. Rep- 
resenting the diameter of the flywheel by D and the lead of the 



Art. 205] 



FRICTION SPINDLE PRESS 



273 



screw by p, both being expressed in inches, the relation existing 
between v s and v t , is 



from which 



h) *• 






(314) 



Let ABC of Fig. 126 represent a part of the curve described by 
the flywheel on the surface of the driving disc; then 

rdd 1 



tan a = 



dr K' 




Fig. 126. 

since the velocity v makes a constant angle with the radius vector. 
Hence, we get by integration 

log e r = KB 

or r = e Ke (315) 

It appears that the curve described by the ffywheel in moving 
across the face of the driving disc is an equiangular or logarithmic 
spiral. 

205. Pressure Developed by a Friction Spindle Press. — (a) 

Working stroke. — Beginning with the ram at the top of the down 
stroke, the friction wheel d being at rest will tend to assume the 
same velocity as the driving disc b, but due to slippage this con- 
dition will not prevail until the screw and flywheel have moved 
downward a certain distance. During the next period the wheel 
d is accelerated with practically no slippage, and when the tool 
strikes the work, the friction wheel, the screw and ram have ac- 



274 FRICTION SPINDLE PRESS [Chap. XI 

cumulated a certain amount of energy which is given out in per- 
forming useful work during the remainder of the stroke. It 
should be noted that the driving disc is thrown out of contact 
with d about the same instant that the tool strikes the work; 
hence the driving force is not considered as doing any useful 
work, but is used merely to accelerate the moving system. 
It is evident, therefore, that the pressure developed during the 
latter part of the stroke depends upon the energy stored up by 
the flywheel, screw, and ram, and the distance through which 
the ram moves in doing the work. 

Assuming that the flywheel d is r 2 inches from the center of 
the driving shaft when the tool strikes the work, the kinetic 
energy in the flywheel and screw due to the tangential velocity 
v t is given by the following expression : 

w W lV f 

in which Wi is the equivalent weight of the wheel and screw re- 
duced to the outside radius of the rim having a velocity of 

7rr 2 iV 

in which N denotes the revolutions per minute made by the 
driving shaft. 

Denoting the actual weight of the wheel, screw and ram by 
the symbol TP 2 , we find that the energy stored up in these parts 
due to the velocity v a is 

in which v s , according to (314), is 

pv t _ pr 2 N 
t^D ~ 360 D 

Now in coming to rest the moving mass Wz also does external 
work, the magnitude of which is 

W 

in which r 3 denotes the distance between the center line of the 
driving shaft and the flywheel at the end of the downstroke. 



Art. 206] EFFICIENCY OF CROWN-FRICTION GEARING 275 

Summing up, we find that the theoretical amount of work that 
ean be done is 

E = E x + E 2 + E 3 (316) 

and multiplying this by the efficiency 77, the external or useful 
work that can be done is -qE. 

The average pressure Q upon the tool multiplied by the dis- 
c's — f~2 

tance — r~ — through which this force acts must be equivalent to 
the work done by the moving system; hence 

Q = J** (317) 

r 3 — r 2 

(b) Return stroke. — On the return stroke, the driving disc c 
is brought into contact with the wheel d, and since the latter is 
at rest for a short interval of time, we have the same conditions 
to contend with that prevailed at the beginning of the working 
stroke, namely, that the flywheel will slip until it attains the 
same speed as the driving disc. After the flywheel has attained 
the speed as the driving disc, this condition will continue until 
the disc c is released and the disc b is again applied. 

206 Double -crown Frictions. — An interesting variable-speed 
friction drive used on the Albany sensitive drill press is shown in 
Fig. 127. It consists of two crown friction wheels, one of which 
is mounted on the drive shaft a, and the other on the spindle k 
of the drill press. A hemisphere c, made of cast iron and bushed 
with bronze, is mounted on a shaft d, which is pivoted on the 
adjustable spindle e. By means of the handle g, the shaft d and 
the hemisphere c may be moved in a vertical plane. The speed 
of rotation of the hemisphere, and the speed of the driven wheel 
h are thus varied. The contact pressure between the hemisphere 
and the friction wheels may be increased or decreased by means 
of the adjusting nut /. Ball bearings are used in all of the 
important bearings on the machine, as shown in Fig. 127. 

207. Efficiency of Crown-friction Gearing. — A study of the 
action of crown-friction gearing shows clearly that the points on 
the disc b in contact with the inner and outer edges of the driven 
wheel c will travel unequal distances per revolution of the disc 
(see Fig. 124). From this it follows that there is slippage be- 



276 



EFFICIENCY OF CROWN-FRICTION GEARING [Chap. XI 



tween the wheel and the disc at the line of contact. Denoting 
by / the width of the face of the wheel c, then the difference 
between the distances traveled per revolution of the disc by the 
extreme points in contact is 2rf. 




To determine the work lost per revolution due to slippage 
multiply the average slip irf by the tangential resistance between 
the wheel and the disc; thus 



W. = irfP 



(318) 



The output per revolution of the disc is /jltPD; hence the total 
work put in, exclusive of that required to overcome the frictional 
resistances of the various bearings, is given by the expression 



W = »irP(D+f) 



(319) 



Art. 208] MOUNTING FRICTION GEARING 277 

Since the efficiency of any machine is equal to the output 
divided by the input, we obtain in this case 

* - 5T7 (320) 

According to (320), the efficiency of crown-friction gearing is 
independent of the diameter of the driven wheel. Furthermore, 
the efficiency increases as the face of the crown wheel is decreased, 
and as the line of contact between the disc and the wheel is moved 
farther from the center of the disc. 



MOUNTING FRICTION GEARING 

In general, friction gearing must be mounted in such a manner 
that the pressure required between the surfaces in contact in 
order to transmit the desired horse power can readily be pro- 
duced. This result is obtained by equipping one of the shafts 
with a special bearing or set of bearings, 

208. Thrust Bearings for Friction Gearing. — (a) Bearings for 
spur and grooved frictions. — In the case of spur and grooved fric- 
tion gearing, the pinion shaft is mounted on eccentric bearings, the 
constructive details of which are shown clearly in Fig. 128. The 
gears themselves should be located close to the bearings in order 
to insure rigidity, thus obviating undue wear on the gears as well 
as on the bearings. 

(b) Thrust bearings for bevel frictions. — For engaging a pair of 
bevel gears, and taking up any wear that may occur, two 
types of bearings are in common use. The first type, which 
may be called a quick-acting end-thrust bearing, is shown in 
Fig. 129. It is used in connection with bevel frictions requiring 
frequent throwing in and out of engagement. The inner sleeve 
a forming the bearing for the shaft has a helical slot into which the 
turned end of the adjusting screw b is fitted. It is evident that 
rotating the sleeve a in the proper direction will cause the sleeve 
to advance in an axial direction, thus engaging the gears. 

The second type of end thrust bearing works on the same general 
principle. The inner sleeve, instead of being fitted with a heli- 
cal slot, is threaded as shown in Fig. 130. This design is well 
adapted to installations in which the friction gears are not en- 
gaged or disengaged very frequently. 



278 



MOUNTING FRICTION GEARING 



rci 



XI 



(c) Thrust bearings for crown frictions. — For engaging crown 
frictions, the same type of bearings as those shown in Figs. 129 












rj 


- 


J 




























L 










-i 


"" I- 





Fig. 128. 



and 130 may be used. Occasionally, spring thrust 
used in place of those just mentioned. 



are 







Fig. 129. 



209. Starting Loads. — As stated in Art. 194, the coefficient of 
friction is a maximum when the slip between the friction gears 





Fig. 130. 



lies between 2 and 6 per cent. Experiments have also shown that 
the coefficient of friction decreases gradually as the slip increases ; 
hence when a friction transmission is started under load, the 



Art. 209] REFERENCES 279 

pressure that must be applied to the surfaces in contact is from 
two to three times as great as that required for normal operation. 
This is due to the decrease in the coefficient of friction caused by 
the excessive slippage during the period of starting. From this 
discussion it follows that the bearings described in the preceding 
paragraphs must be designed for the starting conditions. After 
the transmission is once started the thrusts on the gears may be 
reduced considerably, thus eliminating excessive wear and lost 
work. 

References 

Machine Design, Construction and Drawing, by Spooner. 

Paper Friction Wheels, Trans. A. S. M. E., vol. 18, p. 102. 

Friction Driven Forty-four Foot Pit Lathe, Trans. A. S. M. E., vol. 24, 
p. 243. 

Power Transmission by Friction Driving, Trans. A. S. M. E., vol. 29, p. 
1093. 

Efficiency of Friction Transmission, The Horseless Age, July 6, 1910. 

Friction Transmission, The Rockwood Mfg. Co. 



CHAPTER XII 
SPUR GEARING 

Friction gearing, as has been stated, is not suitable for the 
transmission of large amounts of power, nor where it is desir- 
able that the velocity ratio between the driving and driven mem- 
bers be absolutely positive. For such a transmission it becomes 
necessary to provide the surfaces in contact with grooves and 
projections, thus providing a positive means of rotation. The 
original surfaces of the frictions then become the so-called pitch 
surfaces of the toothed gears, and the projections together with 
the grooves form the teeth. These teeth must be of such a form 
as to satisfy the following conditions: 

(a) The teeth must be capable of transmitting a uniform ve- 
locity ratio. The condition is met if the common normal at 
the point of contact of the tooth profiles passes through the 
pitch point, i.e., the point of tangency of the two pitch lines. 

(6) The relative motion of one tooth upon the other should be 
as much a rolling motion as possible on account of the greater 
friction and wear attendant to sliding. With toothed gearing, 
however, it is impossible to have pure rolling contact and still 
maintain a constant velocity ratio. 

(c) The tooth should conform as nearly as possible to a canti- 
lever beam of uniform strength, and should be symmetrical on 
both sides so that the gear may run in either direction. 

(d) The arc of action should be rather long so that more than 
one pair of teeth may be in mesh at the same time. 

210. Definitions. — Before taking up the discussion of the 
various types of tooth curves, it is well to familiarize ourselves 
with the meaning of different terms and expressions used in con- 
nection with gearing of all kinds. 

(a) By the term circular pitch is meant the distance from one 
tooth to a corresponding point on the next tooth, measured on the 
pitch circle. The circular pitch is equal to the circumference of 
the pitch circle divided by the number of teeth in the gear. 

(b) The diametral pitch is equal to the number of teeth in the 

280 



Art. 210] 



DEFINITIONS 



281 



gear divided by the pitch diameter. It is not a dimension on the 
gear, but is simply a convenient ratio. 

(c) The term chordal pitch may be defined as the distance from 
one tooth to a corresponding point on the next measured on a 
chord of the pitch circle instead of on the circumference. This 
pitch is used only in making the drawing or by the pattern maker 
if the teeth are to be formed on a wood pattern. 

(d) The thickness of the tooth is the thickness measured on the 
pitch line, as illustrated in Fig. 131. 

(e) By the tooth space is meant the width of the space on the 
pitch line. 

(/) The term backlash means the difference between the tooth 
space and the thickness of the tooth. 




ness 



Fig. 131. 



(g) By the term addendum is meant the distance from the pitch 
circle to the ends of the teeth, as dimension a in Fig. 131. 

(h) The distance b between the pitch circle and the bottom of 
the tooth space is called the dedendum. 

(i) The clearance is the difference between the addendum and 
the dedendum, or in other words, the amount of space bstween 
the root of a tooth and the point of the tooth that meshes with it. 

(j) As shown in Fig. 131, the face of the tooth is that part of 
the tooth profile which lies between the pitch circle and the 
end of the tooth. 

(fc) The flank of the tooth is that part of the tooth profile which 
lies between the pitch circle and the root of the tooth, as 
represented in Fig. 131. 

(I) The line of centers is the line passing directly through both 
centers of a pair of mating gears. 



282 



TOOTH CURVES 



[Chap. XII 



(m) The pitch circles of a pair of gears are' imaginary circles, 
the diameters of which are the same as the diameters of a pair of 
friction gears that would replace the spur gears. 

(n) The base circle is an imaginary circle used in involute gear- 
ing to generate the involutes which form the tooth profiles. It 
is drawn tangent to the line representing the tooth thrust, as 
shown in Fig. 131. 

(o) The describing circle is an imaginary circle used in cycloidal 
gearing to generate the epicycloidal and hypocycloidal curves 
which form the tooth profiles. There are two describing circles, 
one inside and one outside of the pitch circle, and they are usu- 
ally of the same size. 

(p) By the angle of obliquity of action is meant the inclination 
of the line of action of the pressure between a pair of mating teeth 
to a line drawn tangent to the pitch circle at the pitch point, as 

represented in Fig. 131 by 
the angle a or the angle 
DCF in Fig. 132. 

(q) The arc of approach is 
the arc measured on the 
pitch circle of a gear from 
the position of the tooth at 
the beginning of contact to 
the central position, that is, 
the arc HC in Fig. 132. 
(r) The arc of recess is the 
arc measured on the pitch circle from the central position of 
the tooth to its position where contact ends, that is, the arc CI 
in Fig. 132. 

(s) The arc of action is the sum of the arcs of approach and 
recess. 

(t) By the term velocity ratio is always meant the ratio of the 
number of revolutions of the driver to the number of revolutions 
of the driven gear. 

211. Tooth Curves. — There are many different types of curves 
that would serve as profiles for teeth and satisfy the condition 
of constant velocity ratio, with sufficient accuracy for all 
practical purposes; but there are in actual use only two, 
namely, the involute and the cycloidal. As regards strength and 
efficiency the two forms are practically on a par. However, the 




Fig. 132. 



Aet. 212] METHODS OF MANUFACTURE 283 

involute tooth has one decided advantage over the cycloidal, 
namely, that the distance between centers may be slightly greater 
or less than the theoretical distance without affecting the velocity 
ratio. The cycloidal tooth, also, has one important advantage 
over the involute, namely, that a convex surface is always in 
contact with one that is concave. Although the contact is 
theoretically a line, practically it is not ; consequently, the wear is 
not so rapid as with involute teeth where the surfaces are either 
convex or straight. 

212. Methods of Manufacture. — Gear teeth are formed in 
practice by two distinct processes, moulding and machine cutting. 
Formerly all gears were cast and the moulds were formed from 
complete patterns of the gears. Of late years, however, gear 
moulding machines are used to a considerable extent, and the 
results obtained are superior to the pattern-moulded gear. 
Even with machine moulding, however, the teeth are somewhat 
rough and warped out of shape, so that the gears always run with 
considerable friction and are not suited to high speeds. At the 
present time gears of ordinary size are almost always cut, except 
those used in the cheaper class of machinery.- The method which 
is commonly used is to cut the teeth with a milling cutter that 
has been formed to the exact shape required. There are also two 
styles of gear planers, one of which generates mathematically 
correct profiles by virtue of the motion given to the cutter and 
the gear blank, and the other forms the outlines by following 
a previously shaped templet. Another method of producing 
machine cut teeth is by the stamping process now used extensively 
in the manufacture of gears for clocks, slot machines, etc. 

A method of generating spur and helical gear teeth by means 
of a hob is now recognized and accepted as the best way of pro- 
ducing accurate teeth. In this generating process a hob, 
threaded to the required pitch, is rotated in conjunction with the 
gear blank at a ratio dependent upon the number of teeth to be 
cut. The cross-section of the thread is a rack that will mesh 
correctly with the gear to be cut. One important advantage of 
this process is that only one hob is required for cutting all num- 
bers of teeth of one given pitch. Another advantage of the nob- 
bing system is that gears can be produced more cheaply than by 
any other system. 



284 INVOLUTE SYSTEM [Chap. XII 

SYSTEMS OF GEAR TEETH 

213. Involute System. — In the involute system of gearing the 
outline of the tooth is an involute of a circle, called the base 
circle. However, when the tooth extends below the base circle 
that portion of the profile is made radial. The simplest concep- 
tion of an involute is as follows: if a cord, which has previously 
been wound around any given plane curve and has a pencil at- 
tached to its free end, is unwound, keeping the cord perfectly 
tight, the pencil will trace the involute of the given curve. The 
base circle may easily be obtained by drawing through the pitch 
point a line making an angle with the tangent to the pitch circle 
at this point, equal to the angle of obliquity of action; then the 
circle drawn tangent to this line will be the required base circle. 

In order to manufacture gears economically, it is essential that 
any gear of a given pitch should work correctly with any other 
gear of the same pitch, thus making an interchangeable set. To 
accomplish this end, standard proportions have been adopted for 
the teeth. 

(a) Angle of obliquity. — The angle of obliquity of action which 
is generally accepted as the standard for cast teeth is 15 degrees, 
although in cases of special design this angle is often made greater. 
When the angle of obliquity is increased, the component of the 
pressure tending to force the gears apart and producing friction 
in the bearings is increased ; but on the other hand, the profile of 
the tooth becomes wider at the base and consequently the 
strength is correspondingly greater. Such gears, having large 
angles of obliquity, are used where the conditions are unusual and 
where the standard tooth form is not suitable. In England, 
teeth of greater obliquity of action and less depth than the stand- 
ard are quite common, and at present there is a tendency in 
that direction in America. For cut teeth now used in motor- 
car construction as well as in machine tools, the manufacturers 
have adopted what is called the stub tooth, having an angle of 
obliquity of 20 degrees. The proportions of the teeth as used 
for this service are given in Art. 223. In designing teeth of the 
stub-tooth form, care must be taken to make the arc of action at 
least as great as the circular pitch; otherwise the teeth would 
not be continuously in mesh and would probably come to- 
gether in such a way as to lock and prevent further rota- 
tion. The standard angle of obliquity of action, adopted by 



Art. 214] 



LAYING OUT THE INVOLUTE TOOTH 



285 



manufacturers of gear cutters and used almost exclusively before 
the advent of the stub teeth, is slightly at variance with the 
usual standard for cast teeth, being 14° 28' 40", the sine of which 
is 0.25. 

(6) Smallest number of teeth. — The smallest involute gear of 
standard proportions that will mesh correctly with a rack of the 
same pitch contains 30 teeth; however, this difficulty is met by 
slightly correcting the points of all the teeth in the set, so that 
a gear of 12 teeth may mesh with any of the other gears of the 
same pitch. The profiles of the teeth may be drawn accurately 
by means of circular arcs having their centers on the base circle 
B, as shown in Fig. 133. The value of these radii for al5-degree 
involute have been carefully worked out by Mr. G. B. Grant of 
the Grant Gear Works and are given in Table 65. 

214. Laying Out the Involute Tooth. — To apply the tabular 
values given in Table 65, draw the pitch, addendum and clearance 
circles in the usual way, and space off the pitch of the teeth on the 
pitch circle. The base circle is constructed next. This may be 
done as described in a preceding 
article or by making the distance a 
in Fig. 133 equal to one-sixtieth of 
the pitch diameter. With the base 
line Bas a circle of centers, draw 
that part of the tooth profile above 
the pitch line A, generally called 
the face of the tooth, by using the 
face radius b given in Table 65. Next draw in that part of the 
tooth profile between the pitch line A and the base circle, using 
the flank radius c given in Table 65. To finish the tooth, that 
part lying between the base circle and the fillet at the root of the 
tooth is made a radial line, as shown in Fig. 133. It should be 
noticed that the values of a, b and c given in Table 65 are for 1 
diametral pitch or 1 inch circular pitch. For any other pitch 
divide or multiply the tabular values by the given pitch as 
directed in the table. 

It will be noted that the tabular values in this table are for 
15-degree involutes and therefore do not apply to the standard 
form of cut teeth. The forms given, however, may be used on 
the drawing, because in cutting a gear the workman needs to 
know only the number of teeth in the gear, and either the number 
of the cutter or the pitch of the hob. All that is required on a 




Fig. 133. 



286 



GRANT'S TABLE FOR INVOLUTE TEETH [Chap. XII 



drawing is an approximate representation of the tooth profile. 
The table also gives values down to a 10-tooth gear, while the 
standard cut gear sets run down to 12 teeth only. This is theo- 
retically the smallest standard involute gear that will have an 
arc of action equal to the circular pitch; however, in the 10- and 
11-tooth gears the error is so slight that it is practically un- 
notioeable. 





Table 


65. — Radii for 15- 


degree Involute Teeth 






ACCOEDING TO 


G. B. Grant 




No. of 
teeth i 


Divide by the 

diametral 

pitch 


Multiply by 

the circular 

pitch 


No. of 
teeth 


Divide by the 

diametral 

pitch 


Multiply by 

the circular 

pitch 


T 


Rad. b 


Rad. c 


Rad. b 


Rad. c 


Rad. 6 


Rad. c 


Rad. b 


Rad. c 


10 


2.28 


0.69 


0.73 


0.22 


28 


3.92 


2.59 


1.25 


0.82 


11 


2.40 


0.83 


0.76 


0.27 


29 


3.99 


2.67 


1.27 


0.85 


12 


2.51 


0.96 


0.80 


0.31 


30 


4.06 


2.76 


1.29 


0.88 


13 


2.62 


1.09 


0.83 


0.34 


31 


4.13 


2.85 


1.31 


0.91 


14 


2.72 


1.22 


0.87 


0.39 


32 


4.20 


2.93 


1.34 


0.93 


15 


2.82 


1.34 


0.90 


0.43 


33 


4.27 


3.01 


1.36 


0.96 


16 


2.92 


1.46 


0.93 


0.47 


34 


4.33 


3.09 


1.38 


0.99 


17 


3.00 


1.58 


0.96 


0.50 


35 


4.39 


3.16 


1.39 


1.01 


18 


3.12 


1.69 


0.99 


0.54 


36 


4.45 


3.23 


1.41 


1.03 


19 


3.22 


1.79 


1.03 


0.57 


37-40 


4.20 


1.34 


20 


3.32 


1.89 


1.06 


0.61 


41-45 


4.63 


1.48 


21 


3.41 


1.98 


1.09 


0.63 


46-51 


5.06 


1.61 


22 


3.49 


2.06 


1.11 


0.66 


52-60 


5.74 


1.83 


23 


3.57 


2.15 


1.13 


0.69 


61-70 


6.52 


2.07 


24 


3.64 


2.24 


1.16 


0.71 


71-90 


7.72 


2.46 


25 


3.71 


2.33 


1.18 


0.74 


91-120 


9.78 


3.11 


26 


3.78 


2.42 


1.20 


0.77 


121-180 


13.38 


4.26 


27 


3.85 


2.50 


1.23 


0.80 


181-360 


21.62 


6.88 



(b) Laying out the rack tooth. — It was found necessary to devise 
a separate means of drafting the rack. The tooth is drawn in the 
usual manner, the sides from the root line to a point midway be- 
tween the pitch and the addendum lines making angles of 75 
degrees with the pitch line. The outer half of the face is formed 
by a circular arc with its center on the pitch line and its radius 
equal to 2.10 inches divided by the diametral pitch, or 0.67 
multiplied by the circular pitch. The radius of the fillet at the 
root of the tooth is taken as one-seventh of the widest part of 
the tooth space. 



Art. 215] INVOLUTE CUTTERS 287 

215. Standard Involute Cutters. — Brown and Sharpe, the 
leading manufacturers of formed gear cutters in this country, 
furnish involute cutters in sets of eight for each pitch, as shown in 
Table 66. 

Table 66. — Brown and Sharpe Standard Involute Cutters 

Cutter No. 1 will cut gears from 135 teeth to a rack. 

Cutter No. 2 will cut gears from 55 teeth to 134 teeth. 

Cutter No. 3 will cut gears from 35 teeth to 54 teeth. 

Cutter No. 4 will cut gears from 26 teeth to 34 teeth. 

Cutter No. 5 will cut gears from 21 teeth to 25 teeth. 

Cutter No. 6 will cut gears from 17 teeth to 20 teeth. 
Cutter No. 7 will cut gears from 14 teeth to 16 teeth. 

Cutter No. 8 will cut gears from 12 teeth to 13 teeth. 

When more accurate tooth forms are desired they also furnish 
to order cutters of the half sizes, making a set of fifteen instead of 
eight cutters. 

All of the above cutters are commonly based on the diametral 
pitch and are made in the following sizes : 

From 1 to 4 diametral pitch, the pitch advances by quarters. 

From 4 to 6 diametral pitch, the pitch advances by halves. 

From 6 to 16 diametral pitch, the pitch advances by whole 
numbers. 

From 16 to 32 diametral pitch, the pitch advances by even 
numbers. 

Then 36, 38, 40, 44, 48, 50, 56, 60, 64, 70, 80, and 120 diametral 
pitch. 

At a slightly greater cost, cutters based on circular pitch may be 
obtained, and the sizes vary as follows: 

From 1 to 1^-inch circular pitch, the pitch advances by 3i$- 
inch increments. 

From \^2 to 3-inch circular pitch, the pitch advances by 34- 
inch increments. 

216. Action of Involute Teeth. — Fig. 134 illustrates the action 
of a pair of involute teeth. Let the circles a and b represent the 
base circles of a pair of involute gears, the pitch circles of which 
would be the circles described about the centers A and B with 
radii of AC and BC respectively. Imagine a cord attached to a 
extending around the circumference to a point D, from there 
directly across to E and around the circumference of b. Let the 
central point of the cord be permanently marked in some manner 



288 



CYCLOIDAL SYSTEM 



[Chap. XII 



and be denoted by C. Now rotate a in the direction of the arrow 
and trace the path of the point C on the surface of a extended, on 
the surface of b extended, and also its actual path in space. It is 
evident that these three curves will be CG, CH, and CJ, and that 
CG and CH will be parts of the involutes of the two base circles 
a and b. Now reverse the rotation of B and rewind the string on 
b until C reaches the point K. During this motion it will com- 
plete the tooth forms CF and FI. Bearing in mind that C is 
always the point of contact of the teeth, its path is evidently JK 
and coincides exactly with the line of pressure between the 
teeth, since the line CD is always normal to the involute curve it 




is generating. If the centers A and B should be misplaced 
slightly on account of wear in the bearings or journals, a uniform 
velocity ratio would still be transmitted because the normals 
would still pass through the point C. The shifting of the centers 
will result in a change of the obliquity of the pressure on the teeth, 
and the length of the arc of contact. The outlines of the teeth 
would not be changed in the least. 

217. Cycloidal System. — The cycloidal system, although the 
oldest, is not so popular as the involute system and seems to be 
gradually going out of use. Mr. Grant in his "Treatise on Gear 
Wheels" says: "There is no more need for two different kinds of 
tooth curves for gears of the same pitch than there is need for 
different kinds of threads for standard screws, or of two different 
kinds of coins of the same value, and the cycloidal tooth would 
never be missed if it were dropped altogether. But it was the 
first in the field, is simple in theory, is easily drawn, has the recom- 
mendation of many well-meaning teachers, and holds its position 



Art. 218] CYLCOIDAL TOOTH FORM 289 

by means of human inertia; or the natural reluctance of the aver- 
age human mind to adopt a change, particularly a change for the 
better." This view is probably a little biased, but nevertheless 
there is a great deal of sound truth in it. The proportion of 
machine cut cycloidal teeth to machine cut involute teeth is very 
small, but in some classes of work, and especially when the loads 
are heavy, the cycloidal forms are still used extensively. 

218. Form of the Cycloidal Tooth. — The outline of a cycloidal 
tooth is made up of two curves. The faces of the teeth are epi- 
cycloids and the flanks are hypocycloids, with two exceptions; 
namely, internal gearing and racks. In the former case, the faces 
are hypocycloids and the flanks are epicycloids, while in the latter 
both curves are plain cycloids. When a circle rolls on a fixed 
straight line, the path generated by an assumed point of the circle 
is a cycloid; should the circle roll on the outside of another circle, 
the path of this point would be an epicycloid, and should it roll 
on the inside of another circle, it would be a hypocycloid. 

These rolling circles are generally spoken of as describing cir- 
cles, and their size determines the form of the tooth, the. arc of 
contact, and the angle of obliquity of action. The angle of ob- 
liquity in the cycloidal system is constantly changing; but its 
average value, when the proportions of the teeth are standard, is 
about 15 degrees, the same as in involute gearing. The circle 
upon which the describing circles are rolled is the pitch circle. 
When the diameter of the rolling circle is equal to the radius of 
the pitch circle, the flanks of the teeth are undercut. In addition 
to the objection that undercut teeth are weak, the amount of 
undercut must be very slight if the teeth are to be cut with a 
rotating cutter. 

The same describing circle must always be used for those parts 
of the teeth which work together, i.e., the faces of a tooth on the 
one gear must be formed by the same describing circle as the 
flanks of the tooth it meshes with. In interchangeable sets it is 
desirable to use the same size describing circle for both the faces 
and the flanks of all the gears of the same pitch, and the size of 
the describing circle which is generally accepted as standard is 
one whose diameter is equal to the radius of a 12-tooth gear of the 
same pitch. Here again, the manufacturers of gear cutters are 
at variance, and use a 15-tooth gear as the base of the system. 
This does not mean that the 15-tooth gear is the smallest gear in 



290 



GRANT'S TABLE FOB CYCLOIDAL TEETH [Chap. XII 



the set, but simply means that smaller gears will have undercut 
flanks. 

219. Laying out the Cycloidal Tooth. — The profiles of cycloidal 
teeth, as in the case of involute teeth, may be very accurately 
represented by circular arcs. In Table 67 are given the radii of 
these arcs, also the radial distances from their centers to the pitch 
line as determined by Mr. Grant. In laying out the profiles of 




Fig. 135. 

cycloidal teeth, draw the pitch, addendum and clearance circles, 
and space off the pitch of the teeth on the pitch circle. Next 
draw the circle B as shown in Fig. 135 at a distance a inside of the 
pitch circle A, also the circle C at a distance e outside of the pitch 
line. The former is the circle of face centers and the latter, the 



Table 67. — Radii foe 


Cycloidal Teeth According to 


G. B'. Grant 


Number of teeth 


Divide by the diametral pitch 


Multiply by the circular pitch 


Exact 


Approx. 


Rad. b 


Dist. o 


Rad. c 


Dist. e 


Rad. b 


Dist. a 


Rad. c 


Dist. e 


10 


10 


1.99 


0.02 


-8.00 


4.00 


0.62 


0.01 


-2.55 


1.27 


11 


11 


2.00 


0.04 


-11.05 


6.50 


0.63 


0.01 


-3.34 


2.07 


12 


12 


2.01 


0.06 


00 


00 


0.64 


0.02 


CO 


CO 


13H 


13-14 


2.04 


0.07 


15.10 


9.43 


0.65 


0.02 


4.80 


3.00 


15^ 


15-16 


2.10 


0.09 


7.86 


3.46 


0.67 


0.03 


2.50 


1.10 


17H 


17-18 


2.14 


0.11 


6.13 


2.20 


0.68 


0.04 


1.95 


0.70 


20 


19-21 


2.20 


0.13 


5.12 


1.57 


0.70 


0.04 


1.63 


0.50 


23 


22-24 


2.26 


0.15 


4.50 


1.13 


0.72 


0.05 


1.43 


0.36 


21 


25-29 


2.33 


0.16 


4.10 


0.96 


0.74 


0.05 


1.30 


0.29 


33 


30-36 


2.40 


0.19 


3.80 


0.72 


0.76 


0.06 


1.20 


0.23 


42 


37-48 


2.48 


0.22 


3.52 


0.63 


0.79 


0.07 


1.12 


0.20 


58 


49-72 


2.60 


0.25 


3.33 


0.54 


0.83 


0.08 


1.06 


0.17 


97 


73-144 


2.83 


0.28 


3.14 


0.44 


0.90 


0.09 


1.00 


0.14 


290 


145-300 


2.92 


0.31 


3.00 


0.38 


0.93 


0.10 


0.95 


0.12 


Ra 


ck 


2.96 


0.34 


2.96 


0.34 


0.94 


0.11 


0.94 


0.11 



Art. 220] CYCLOID AL CUTTERS 291 

circle of flank centers. The tooth profile may now be drawn us- 
ing the face and flank radii b and c given in Table 67 for the num- 
ber of teeth to be used in the gear. The values given for a, b, 
c, and e in Table 67 are for diametral pitch or 1 inch circular pitch. 
For any other pitch, divide or multiply the tabulated values by 
the given pitch as directed in the table. 

The smallest gear in the set is again one having ten teeth, while 
the smallest one for which standard cutters are manufactured is 
one having 12 teeth. The tooth form obtained by using the tabu- 
lar values as directed above differs slightly from that obtained by 
the use of standard cutters on account of the difference in the 
describing circles, but as in the case of involutes, the discrepancy 
is small and for that reason Grant's tabular values may be used 
for representing the tooth form on a drawing. 

220. Standard Cycloidal Cutters. — The Brown and Sharpe 
Mfg. Co. furnish sets of cycloidal cutters based on the diametral 
pitch only, and the sizes vary as follows: 

From 2 to 3 diametral pitch, the pitch varies by quarters. 

From 3 to 4 diametral pitch, the pitch varies by halves. 

From 4 to 10 diametral pitch, the pitch varies by whole num- 
bers. 

From 10 to 16 diametral pitch, the pitch varies by even num- 
bers. 

Each set consists of 24 cutters, as indicated in Table 68. 

Table 68. — Brown and Sharpe Standard Cycloidal Cutters 
Cutter A for gears having 12 teeth. 
Cutter B for gears having 13 teeth. 
Cutter C for gears having 14 teeth. 
Cutter D for gears having 15 teeth. 
Cutter E for gears having 16 teeth. 
Cutter F for gears having 17 teeth. 
Cutter G for gears having 18 teeth. 
Cutter H for gears having 19 teeth. 
Cutter I for gears having 20 teeth. 
Cutter J for gears having 21 to 22 teeth. 
Cutter K for gears having 23 to 24 teeth. 
Cutter L for gears having 25 to 26 teeth. 
Cutter M for gears having 27 to 29 teeth. 
Cutter N for gears having 30 to 33 teeth. 
Cutter O for gears having 34 to 37 teeth. 
Cutter P for gears having 38 to 42 teeth. 
Cutter Q for gears having 43 to 49 teeth. 
Cutter R for gears having 50 to 59 teeth. 



292 



ACTION OF CYCLOIDAL TEETH 



[Chap. XII 



Table 68. — Brown and Sharpe Standard Cycloidal Cutters. — 
(Continued.) 
Cutter S for gears having 60 to 74 teeth. 
Cutter T for gears having 75 to 99 teeth. 
Cutter U for gears having 100 to 149 teeth. 
Cutter V for gears having 150 to 249 teeth. 
Cutter W for gears having 250 or more. 
Cutter X for gears having rack. 

221. Action of Cycloidal Teeth. — The action of a pair of cy- 
cloidal teeth is illustrated in Fig. 136. Let the circles a and b 
represent the pitch circles of a pair of gears having cycloidal 
teeth, and let the circles d and e represent the desc ibing circles. 
Let C be the pitch point, and d and C e be the points on the circles 
d and e which coincide with C when the teeth are in the position 
shown in the figure. Now let the centers of the circles a, b, d, 
and e be fixed and rotate a in the direction indicated by the arrow. 




Fig. 136. 



Let the contact at C be so arranged that the circles b, d, and e are 
driven with the same peripheral speed as a. Trace the path of 
the point Ca on the surface of a extended, on the surface of b 
extended, and also its actual path in space. These paths will 
evidently be the hypocycloidal flank CF, the epicycloidal face 
CH of the meshing tooth, and the path of the point of contact 
CJ. Now replace the mechanism in its original position, rotate 
a in the opposite direction and trace the path of C e in the same 
manner. The curves CG, CI and CK, are thus formed and they 
complete the two tooth forms and the path of contact. As the 
line of pressure between the teeth, which of course coincides with 
the common normal at the point of contact, must always pass 



Aet. 221] 



STRENGTH OF SPUR GEARING 



293 



through the point C in order to transmit a uniform velocity, the 
angle of obliquity varies from the angle JCL to zero during the 
arc of approach, and from zero to the angle KCM, which equals 
the angle JCL, during the arc of recess. In order to show that 
with this form of tooth the normal to the tooth profile at the 
point of contact always passes through the pitch point C, let us 
study Fig. 137. It is evident that the generating point C e , 
as well as every other point on the rolling circle, is at any 
given instant rotating about the point 
of contact C of the rolling circle 
with the pitch line. Therefore, at 
the instant in question the line CC e 
is a radius for the point C e and is con- 
sequently normal at that point to 
the curve which C e is generating. 
Now referring again to Fig. 136, the 
point at which the rolling circle is 
always in contact with the pitch 
circle is evidently the pitch point, 
and therefore the common normal 
at the point of contact always 
passes through it. 




Fig. 137. 



STRENGTH OF SPUR GEARING 

Having determined the proper form of a gear tooth, the next 
step is to determine its proportions for strength. Owing to the 
inaccuracy of forming and spacing the teeth, it is customary to 
provide sufficient strength for transmitting the entire load by one 
tooth, rather than considering the load as distributed over the 
whole number of teeth in theoretical contact. 

The load on a single tooth, when the gears are cast from wood 
patterns, is often concentrated at some one point, usually an outer 
corner, on account of the draft on the teeth and the natural warp 
of the castings. The same result is liable to be produced when 
the shaft is weak or when the gears are not supported on a rigid 
framework or foundation. However, in the case of well-sup- 
ported machine-moulded or cut gears, the load may be considered 
as uniformly distributed along the tooth. For the reason just 
stated, the subject of the strength of teeth will be discussed under 
two heads as follows: (a) strength of cast teeth; (b) strength of cut 
teeth. 



294 



STRENGTH OF CAST TEETH 



[Chap. XII 



222. Strength of Cast Teeth. — In deriving the formula for the 
maximum load that a gear with cast teeth will transmit, it will 
be sufficiently accurate to consider the shape of the tooth as 
rectangular, and the load as acting at the outer end. The load 
may, however, be concentrated at one corner or uniformly 
distributed along the length of the tooth. 

(a) Load at one corner. — With the load concentrated at an 
outer corner as shown in Fig. 138, it is probable that rupture 
would occur along a section making some angle a with the base 
of the tooth. Equating the bending moment about the critical 

section due to W to the resist- 
ing' moment of the section, 
we have 

Sht 2 




Whcos a 



6 sin 



in which S denotes the allow- 
able working stress in the 
material. From this we get 

3TFsin 2a 



S = 



(321) 



Fig. 138. 



The stress S is maximum 
when sin2a is maximum, or 
when a is equal to 45 degrees; therefore, 

SW 

t 2 



Max. S = 



(322) 



(b) Load uniformly distributed. — When the load is uniformly 
distributed along the length of the tooth, we have by equating the 
bending moment at the base of the tooth to the resisting moment, 



from which 



Wh = 



S = 



SfP 
6 ' 

ft 2 



(323) 



(c) Equal strength. — Assuming that a tooth is equally strong 
against both methods of failure, the relation existing between 
the height h and the face / is found by equating the stresses given 
by (322) and (323). Hence 



Art. 222] STRENGTH OF CAST TEETH 295 

/ = 2 h = 1.4 p', (324) 

where h = 0.7 p' and p' denotes the circular pitch of the gear. 

Although, as shown by (324), the theoretical length of face at 
which the teeth will be of equal strength for both cases of loading 
is 1.4 p', a well-known American engineer, C. W. Hunt, taking 
his data from actual failures in his own work, states that the face 
should be about 2 p' in order to satisfy this condition. 

The seeming discrepancy between theory and actual results 
may be easily explained when one takes into consideration the 
fact that even if the load may be entirely concentrated at the 
corner at the beginning of application of the load, it is very pro- 
bable that before the full pressure is brought to bear a slight de- 
flection of the outer corner will cause the load to be distributed 
along a considerable length of the face. Another condition 
which adds to the length of the face is that of the proper propor- 
tions for wearing qualities, and in some cases the faces are made 
extra long for that purpose alone. It is customary in American 
practice to make the face of cast teeth two to three times the cir- 
cular pitch, the length of the face being increased as the quality 
of the work is improved. 

(d) Common proportions of cast teeth. — The proportions of cast 
gear teeth as used by the different manufacturers of transmission 
gears vary somewhat, but for ordinary service the following 
proportions in terms of the circular pitch have proven satisfac- 
tory in actual practice : 

Pressure angle or angle of obliquity = 15 degrees. 
Length of the addendum = 0.3 p'. 
Length of the dedendum = 0.4 p'. 
Whole depth of the tooth = 0.7 p'. 
Working depth of the tooth = 0.6 p'. 
Clearance of the tooth = 0.1 p' 
Width of the tooth space = 0.525 p'. 
Thickness of the tooth = 0.475 p'. 
Backlash = 0.05 p'. 

(e) Allowable working load for cast teeth. — Assuming the pro- 
portions of the teeth as given above, we find from (323) that the 
allowable working load on cast gear teeth has a magnitude given 
by the following expression : 

W = 0.054 Sp'f (325) 



296 



STRENGTH OF CUT TEETH 



[Chap. XII 



This formula has the same general form as the well-known 
Lewis formula given in Art. 223. The magnitude of the safe 
working stress depends upon the material, the class of service, 
and the speed at which the gears are operated. If the gears are 
subjected to heavy shocks, due allowance must be made for such 
shocks. To obtain the probable safe working stress for a given 
speed and material, use (330) and Table 72. 

223. Strength of Cut Teeth.— In 1893, Mr. Wilfred Lewis pre- 
sented at a meeting of the Engineers' Club of Philadelphia an 
excellent method of calculating the strength of cut gear teeth. 
His investigation was the first one to take into consideration the 
form of the tooth profile and the fact 
that the direction of the pressure is 
always normal to the tooth profile. 
The Lewis method has since that time 
been almost universally adopted for 
calculating the strength of teeth when 
the workmanship is of high grade, as in 
the cut gears, and not infrequently 
for machine-moulded teeth. 

In this investigation, Mr. Lewis as- 
sumed that at the beginning of contact 
the load was concentrated at the end of 
the tooth, with its line of action normal 
to the tooth profile in the direction AB 
as shown in Fig. 139. The actual thrust P was then resolved at 
the point B into two components, one acting radially producing 
pure compression, and the other, W, acting tangentially. When 
the material of which the gears are made is stronger in compres- 
sion than in tension, the radial component adds to the strength 
of the tooth, and when the tensile and compressive strengths are 
approximately equal, it is a source of weakness. However, in 
either case the effect is not marked, and in the original investiga- 
tion was neglected altogether. 

The strength of the tooth may now be determined by drawing 
through the point B, Fig. 139, a parabola which is tangenj; to 
the tooth profile at the points D and E. This parabola then en- 
closes a cantilever beam of uniform strength as the following 
analysis shows. 

A beam of uniform strength is one in which the fiber stress due 
to bending is constant. For the case under discussion, by equat- 



''/ 


x ^ \ 


/ 


\ \ 
\ \ 

\\ 




H ' 


E 




T ' 




\ 


G 


K-' 









Fig. 139. 



Art. 223] 



TABLE OF LEWIS FACTORS 



297 



ing the external moment to the moment of resistance, we obtain 



Wh 



Sft* 



from which 



t 2 = 



QWh 
Sf 



Kh: 



(32(3) 



(327) 



thus proving that a beam of uniform strength has a parabolic 
outline. 

Since the actual tooth and the inscribed parabola have the 
same value of t as shown in Fig. 139, it is evident that the para- 
bolic beam must be a measure of the strength of the gear tooth, 
and that the weakest section of the tooth must lie along DE. 

The problem now is to find an expression for the load W in 
terms of the dimensions of the tooth, the safe fiber stress and a 
constant. From the similar triangles shown in Fig. 139, it fol- 
lows that 

t 2 = 4hx (328) 

Combining (326) and (328), we find 



W 



Sfx 







Table 69. — Lewis Factors for Gearing 






No. of 


Involute 


Radial 
flank 


Cycloid 


No. of 
teeth 


Involute 


Radial 
flank 


Cycloid 


teeth 


15° 


20° 


15° 


20° 


12 


0.067 


0.0780 


0.0520 




40 


0.1070 


0.1312 


0.0674 




13 


0.071 


0.0840 


0.0530 




45 


0.1080 


0.1340 


0.0682 




14 


0.075 


0.0890 


0.0540 




50 


0.1100 


0.1360 


0.0690 




15 


0.078 


0.0930 


0.0550 




55 


0.1120 


0.1375 






16 


0.081 


0.0970 


0.0560 




60 


0.1130 


0.1390 


0.0700 




17 


0.084 


0.1000 


0.0570 


Same 


65 


0.1140 


0.1400 




Same 


18 


0.086 


0.1030 


0.0580 


values 


70 


0.1144 


0.1410 




values 


19 


0.088 : 0. 1060 


0.0590 


as 


75 


0.1150 


0.1420 


0.0710 


as 


20 


0.090 


0.1080 


0.0600 


for 


80 


0.1155 


0.1426 




for 


21 


0.092 


0.1110 


0.0610 


15° 


90 


0.1164 


0.1440 




15° 


22 


0.093 


0.1130 


0.0615 


invo- 


100 


0.1170 


0.1450 


0.0720 


invo- 


23 


0.094 


0.1140 


0.0620 


lute 


120 


0.1180 


0.1460 




lute 


24 


0.096 


0.1160 


0.0625 




140 


0.1190 


0.1475 






26 


0.098 


0.1190 


0.0635 




160 


0.1197 


0.1483 






28 


0.100 


0.1220 


0.0643 




180 


0.1202 


0.1490 






30 


0.101 


0.1240 


0.0650 




200 


0.1206 


0.1495 


0.0730 




33 


0.103 


0.1260 


0.0657 




250 


0.1213 


0.1504 






36 


0.105 


0.1290 


0.0665 




300 


0.1217 


0.1510 


0.0740 




39 


0.107 


0.1306 


0.0672 




Rack 


0.1240 


0.1540 


0.0750 





298 



LEWIS FACTORS FOR STUB-TEETH [Chap. XII 



Dividing and multiplying by p', the circular pitch, 

W = Sp'fy, (329) 

2 x 
in which y = ~ — > is a factor depending upon the pitch and form 

6 V 
of the tooth profile. The value of this factor must be obtained 
from a layout of the tooth, provided a table of such factors is not 
available. For convenience, the factor y will hereafter be known 
as the "Lewis factor" and in Table 69 are given the values of this 



Table 70. — Values op y in 


Lewis 


' Formula for Stub-tooth 


Gears. 


No. of 
teeth 


Fellows system 


Nuttall 
system 


% 


W 


% 


K 


Ho 


Hi 


i0 Az 


W* 


12 


0.096 


0.111 


0.102 


0.100 


0.096 


0.100 


0.093 


0.092 


0.099 


13 


0.101 


0.115 


0.107 


0.106 


0.101 


0.104 


0.098 


0.096 


0.103 


14 


0.105 


0.119 


0.112 


0.111 


0.106 


0.108 


0.102 


0.100 


0.108 


15 


0.108 


0.123 


0.115 


0.115 


0.110 


0.111 


0.105 


0.103 


0.111 


16 


0.111 


0.126 


0.119 


0.118 


0.113 


0.114 


0.109 


0.106 


0.115 


17 


0.114 


0.129 


0.122 


0.121 


0.116 


0.116 


0.111 


0.109 


0.117 


18 


0.117 


0.131 


0.124 


0.124 


0.119 


0.119 


0.114 


0.111 


0.120 


19 


0.119 


0.133 


0.127 


0.127 


0.122 


0.121 


0.116 


0.113 


0.123 


20 


0.121 


0.135 


0.129 


0.129 


0.124 


0.123 


0.118 


0.115 


0.125 


21 


0.123 


0.137 


0.131 


0.131 


0.126 


0.125 


0.120 


0.117 


0.127 


22 


0.125 


0.139 


0.133 


0.133 


0.128 


0.126 


0.122 


0.118 


0.128 


23 


0.126 


0.141 


0.134 


0.135 


0.129 


0.128 


0.123 


0.120 


0.130 


24 


0.128 


0.142 


0.136 


0.136 


0.131 


0.129 


0.125 


0.121 


0.131 


25 


0.129 


0.143 


0.137 


0.138 


0.133 


0.130 


0.126 


0.123 


0.133 


26 


0.130 


0.145 


0.139 


0.139 


0.134 


0.132 


0.128 


0.124 


0.134 


27 


0.132 


0.146 


0.140 


0.140 


0.135 


0.133 


0.129 


0.125 


0.136 


28 


0.133 


0.147 


0.141 


0.141 


0.136 


0.134 


0.130 


0.126 


0.137 


29 


0.134 


0.148 


0.142 


0.143 


0.137 


0.135 


0.131 


0.127 


0.138 


30 


0.135 


0.149 


0.143 


0.144 


0.138 


0.136 


0.132 


0.128 


0.139 


32 


0.137 


0.150 


0.145 


0.146 


0.140 


0.137 


0.134 


0.130 


0.141 


35 


0.139 


0.153 


0.147 


0.148 


0.143 


0.139 


0.136 


0.132 


0.143 


37 


0.140 


0.154 


0.149 


0.149 


0.144 


0.141 


0.138 


0.133 


0.145 


40 


0.142 


0.156 


0.151 


0.151 


0.146 


0.142 


0.140 


0.135 


0.146 


45 


0.145 


0.159 


0.154 


0.154 


0.149 


0.145 


0.142 


0.138 


0.149 


50 


0.147 


0.161 


0.156 


0.156 


0.151 


0.147 


0.144 


0.140 


0.151 


55 


0.149 


0.162 


0.157 


0.158 


0.152 


0.149 


0.146 


0.141 


0.153 


60 


0.150 


0.164 


0.159 


0.159 


0.154 


0.150 


0.148 


0.143 


0.154 


70 


0.153 


0.166 


0.161 


0.161 


0.156 


0.152 


0.150 


0.145 


0.157 


80 


0.155 


0.168 


0.163 


0.163 


01.58 


0.154 


0.152 


0.147 


0.159 


100 


0.158 


0.171 


0.166 


0.166 


0.160 


0.156 


0.154 


0.150 


0.161 


150 


0.162 


0.174 


0.170 


0.169 


0.164 


0.160 


0.158 


0.154 


0.165 


200 


0.164 


0.176 


0.172 


0.171 


0.166 


0.162 


0.160 


0.156 


0.167 


Rack 


0.173 


0.184 


0.179 


0.176 


0.172 0.170 


0.168 


0.166 


0.175 



Art. 224] 



PROPORTIONS OF CUT TEETH 



299 



factor as worked out by Mr. Lewis for the several systems of gear- 
ing. In Table 70 are given the values of the Lewis factor 
for the two systems of stub-tooth gearing in common use. 
These factors were derived and tabulated by Mr. L. G. Smith 
under the direction of the author, and formed a part of a thesis 
submitted by Mr. Smith. 

(b) Proportions of cut teeth. — The proportions of cut teeth as 
recommended by several manufacturers of gear-cutting machin- 
ery vary considerably, as may be noticed from an inspection of the 
formulas given in Table 71. No doubt the formulas proposed 
by the Brown and Sharpe Co. for the common system of gearing 
are used more extensively than any other and are generally 
recognized as the standard. The formulas due to Hunt apply to 
short teeth, while those proposed by Messrs. Logue and Fellows 
apply to the well-known stub systems of gear teeth. No for- 
mulas are given in Table 71 for the Fellows stub teeth since this 
system is discussed more in detail in Art. 230 (d). It should be 
noted that the proportions recommended by Messrs. Hunt and 
Logue agree on all points except the pressure angle. 



Table 71. — Proportions of Cut Teeth 





Brown and 
Sharpe 


Hunt 


Logue 


Fellows 


Pressure angle 


14^° 
0.3183 p' 
0.3683 p' 
0.6866 p' 
. 6366 p' 


14M° 
0.25 p' 
0.30 p' 
0.55 p' 
50 r>' 


20° 
0.25 p' 
0.30 p' 
0.55 p' 
50 t)' 


20° 














Clearance 

Width of tooth space 

Thickness of the tooth 


0.05 p' 
0.50 p' 
. 50 ?)' 




Backlash 












Another important fact shown in the table is that for cut teeth 
the backlash is zero. 

224. Materials and Safe Working Stresses. — (a) Materials 
used in gears. — The factor S in the Lewis formula depends upon 
the material used in the construction of the gears. The materials 
used for gear teeth are various grades of alloy steels, machine 
steel, steel casting, semi-steel, cast iron, bronze, rawhide, cloth, 
fiber, and wood. Machine-steel pinions are used with large cast- 
iron gears; the use of the stronger material makes up for the 



300 MATERIALS FOR GEARS [Chap. XII 

weakness of the teeth on the pinion, due to the decreased section 
at the root. At the present time the majority of the gears used 
in motor-car construction are made of steel and are then sub- 
jected to a heat treatment, the effect of which has been discussed 
in Arts. 52 and 53. Many gears on modern machine tools and 
electric railway cars are made of steel and then given a heat 
treatment. Steel gears heat treated are stronger and are 
capable of resisting wear much better than untreated gears. 

Steel casting is used when the gears are of large size. This 
material is well adapted for resisting shocks and, being much 
stronger than cast iron, it is used for service in which heavy loads 
prevail. Semi-steel, which is nothing more than a high-grade 
cast iron, is also used for large gears where the shocks and loads 
are not so severe. Cast iron probably is used more frequently 
than any other material, and in many cases the manufacturers 
of gears use a special cupola mixture that will produce a tough 
and close-grained metal. 

Bronze is frequently used for spur pinions meshing with steel 
or iron gears, and when the teeth are properly cut the gears 
may be run at fairly high speeds. In worm-gear installations, 
the gear is generally made of bronze and the worm of a good grade 
of steel, in many cases heat treated. Several manufacturers are 
now making special gear bronzes that are adapted for a particular 
type of service. Some of these bronzes are discussed more or less 
fully in the chapter on worm gearing. In general, bronze is 
much stronger than ordinary cast iron when applied to gear teeth. 

Rawhide, cloth, and fiber gears are used when quiet and smooth- 
running gears, free from vibration, are desired. Rawhide gears 
are stronger and are preferable to ordinary fiber ones. The 
New Process Rawhide Co. claims its gears to be equally as strong 
as cast-iron gears of the same dimensions. Such gears are fur- 
nished with or without metal flanges and bushings, and the teeth 
are cut the same as in a metal gear. As ordinarily constructed, 
the flanges and hub of the smaller gears are made of brass or 
bronze and for the larger ones cast iron or steel may be used. 
In the case of large gears only the teeth and rim are of rawhide, 
the center being of cast iron. As a rule, however, rawhide gears 
are of small size. They are often used as the driving pinions 
on motor drives, and the fact that rawhide is a non-conductor is 
in this service a marked advantage. 

The cloth or so-called "Fabroil" gears introduced several 



Art. 224] SAFE WORKING STRESS 301 

years ago by the General Electric Co. consist of a filler of cotton 
or similar material confined at a high pressure between steel 
flanges held together by either threaded rivets or sleeves, depend- 
ing upon the size of the gears. After cutting the teeth in the 
blank, the gear is subjected to an oil treatment making it mois- 
ture-proof as well as vermin-proof. In strength, Fabroil gears 
are the equal of other non-metallic gears, and according to the 
manufacturer they may be used in practically any service where 
cast iron gears are used. 

Recently the Westinghouse Electric and Manufacturing Co. 
placed upon the market a non-metallic or fibrous material, called 
Bakelite Micarta-D, that is suitable for gears and pinions. It 
is especially adapted for installations where it is desirable to 
transmit power with a minimum amount of noise. This material 
possesses good wearing qualities, is vermin-proof, absorbs practi- 
cally no oil or water, and is unaffected by atmospheric changes 
and acid fumes. Furthermore, gears made of this material may 
be run in oil without showing any signs of injury; in fact, the 
manufacturers specify that a good lubricating oil or grease is 
essential in order to obtain good results. According to recorded 
tests, the ultimate tensile strength of Bakelite Micarta-D is 
approximately 18,000 pounds per square inch with the grain, 
while its compressive strength across the grain is 40,000 pounds 
per square inch. 

(b) Safe working stress. — The factor S in (329) depends upon 
the kind of material used, the conditions under which the gears 
run and the velocity of the gears. If the gears are subjected to 
severe fluctuations of load or to shock, or both, due allowance 
must be made. To provide against the effect of speed, Mr. 
Lewis published a table of allowable working stresses for a few 
types of materials. Some years later Mr. C. G. Barth originated 
an equation giving values for S which agree very closely with 
those recommended by Mr. Lewis. The Barth formula is gener- 
ally put into the following form : 

s - [sbtt] s °- < 330 > 

in which So denotes the permissible fiber stress of the material at 
zero speed and V the pitch line velocity in feet per minute. In 
Table 72 are given values of So for the various materials 
discussed. 



302 TABLE OF VALUES S [Chap. XII 

Table 72. — Values op So for Various Materials 



So 



Chrome nickel steel, hardened 
Chrome vanadium steel, hardened 

Alloy steel, case-hardened 

Machinery steel 

Steel casting 

Special high-grade bronze 

Ordinary bronze 

High-grade cast iron (semi-steel). . 

Good cast iron 

Ordinary cast iron 

Fabroil 

Bakelite Micarta-D 

Rawhide 



100,000 

100,000 

50,000 

25,000 

20,000 

16,000 

12,000 

15,000 

10,000 

8,000 

8,000 

8,000 

8,000 



GEAR CONSTRUCTION 

The constructive details of gears depend largely upon the size, 
and to some extent upon the material used as well as upon the 
machine part to which the gears are fastened. Small metal 
gears are generally made solid, but when the diameter gets too 
large for this type of construction thus producing a heavy gear, 
the weight of such gears can be materially decreased by recessing 
the sides thus forming a central web connection between the rim 
and the hub. Not infrequently round holes are put through the 
web, thus effecting an additional saving in weight and at the same 
time giving the gear an appearance of having arms. Gear blanks 
having a central web are usually produced by casting, or by a 
drop forging operation. 

225. Rawhide Gears. — Rawhide gears, as mentioned in the pre- 
ceding article, are always provided with metal flanges at the side 
as illustrated in the various designs shown in Figs. 140 and 141. 
For spur gears up to and including 9 inches outside diameter, the 
metal flanges are fastened together by means of rivets with coun- 
tersunk heads as shown. For larger outside diameters either 
rivets or through bolts are used, depending largely upon sur- 
rounding conditions. 

The design shown in Fig. 140(a), having the plates extending 
almost to the roots of the teeth, produces a very quiet running 
gear which gives good service for light and medium loads. In this 



Art. 225] 



RAWHIDE GEARS 



303 



case the flanges are merely used for supporting the key If a 
stronger rawhide gear is desired than that just described, the 
flanges must be extended to the ends of the teeth, thus forming the 
combination shown in Fig. 140(6). The flanges may or may not 







(a) 






: . 



7%m 



(b) 

Fig. 140. 




(c) 



form a part of the working face. If the working face does not 
include the flanges, the rawhide filler must be made Y± inch wider 
than the face of the engaging gear; furthermore, if this gear is 
used as a motor pinion, the rawhide face must be considerably 





(b) 



Fig. 141. 



wider than the face of the mating gear in order to compensate 
for the floating of the armature shaft. The object of extending 
the flanges to the tops of the teeth is to prevent the outer layers 
of rawhide from curling over and thus eventually ruining the 
whole gear. 



304 



RAWHIDE GEARS 



[Chap. XII 



The design shown in Fig. 140(c) is intended for severe service. 
Quiet operation is obtained by eliminating the metal to metal 
contact, and this is accomplished by making the rawhide face 
somewhat wider than the face of the engaging gear. The con- 
struction shown in Fig. 141 (a) is that used for large gears, thus 
effecting a considerable saving of rawhide by using the cast-iron 
spider to which the rawhide rim is fastened as shown. The 
flanges may or may not extend to the tops of the teeth. When 
the face of such a gear is 4 inches or more, through bolts are 
generally used in place of rivets, unless the projecting heads and 
nuts are found objectionable. 

For the constructions shown in Fig. 140(a) and (6), the thick- 
ness of the plates may be made according to the dimensions given 
in Table 73. This table also gives the size of rivets to be used 
for a given pitch of tooth and for the ordinary length of face, 
namely, about three times the circular pitch. The last two col- 
umns given in the table refer to the minimum radial thickness of 
the rawhide blank when used without and with a metal spider. 

Table 73. — Data Pertaining to Rawhide Gears 





Flange 
thickness 


Diameter of 
rivet 


Thickness of rawhide rim 


Diametral pitch 


Without metal 
spider 


With metal 
spider 


12 


y 8 


%2 


0.445 
0.550 
0.590 
0.640 
0.725 
0.890 
1.100 
1.275 
1 . 675 
1.780 
1.905 
2.140 
2.330 


0.415 


10 


y s 


0.505 


9 




0.545 


8 


H 


0.590 


7 


%4 


0.670 


6 


He 


0.800 


5 

4 


H 


0.975 
1.150 


3 


% 


Yx& 


1.500 


2% 




1.610 


2% 


V2 


1.735 


2H 
2 


% 


1.980 
2.175 



The information included in Table 73 was kindly furnished by 
the New Process Gear Corporation and represents their practice 
in the ordinary designs of rawhide gears. 



Art. 226] 



FABROIL GEARS 



.305 



226. Fabroil Gears. — The general constructive features of 
Fabroil gears are very much the same as those used for rawhide 
gears. In the usual construction as recommended by the General 
Electric Co., the flanges are made of steel and threaded studs are 



7 r 



!■ 



1 



i 



^wmm 



ran 




(a) 



(b) 



used for clamping the flanges together. Four different designs 
of such gears are shown in Figs. 142 and 143. The first of these, 
Fig. 142(a), shows the standard construction without a metal 
bushing or spider. For gears of a larger size, a flanged bushing 






(a) 



(b) 



made of machine steel or steel casting, depending upon the size 
of the gear, is employed, as shown in Fig. 142(6). This form of 
construction is used for the sake of economy of material. 
The two flanges are locked together by the threaded studs, 



306 BAKELITE MICARTA-D GEARS [Chap. XII 

and in addition the removable flange is locked to the bushing by 
several studs tapped half into the flange and half into the cen- 
tral bushing. 

The design shown in Fig. 143(a) represents the construction 
used for large gears when it is desirable to save material. The 
Fabroil rim is pressed over a metal spider and locked in place 
by the system of threaded studs shown in the figure. The 
threaded sleeve construction shown in Fig. 143 (o) is used for 
small pinions where there is not sufficient room for the threaded 
studs used in the standard construction. The end flanges are 
locked to the threaded sleeve in the manner shown in the figure. 

227. Bakelite Micarta-D Gears. — For ordinary service, in 
which the face of the Bakelite Micarta-D is made equal to or less 
than the face of the mating gear, no end flanges are required since 
the material is self-supporting. However, end flanges are recom- 
mended when it is desired to transmit heavy loads or when the 
diameter of the gear is more than four times its face. Bakelite 
Micarta-D is obtainable in the form of plates up to 36 inches 
square and in thicknesses varying from }/{§ inch to 2 inches; 
hence the largest gears that can be made are limited to 36 inches 
outside diameter, but the face may be made any width whatsoever 
by riveting together two or more plates. For economy of mate- 
rial, large-diameter gears are made with a metal center similar to 
that of the Fabroil gear shown in Fig. 143(a). 

228. Large Gears. — Gears of medium diameter are cast in one 
piece, either of cast iron or of steel casting depending upon the 
class of service for which they are intended. Quite often the 
gear is cast in one piece and in order to relieve the shrinkage 
stresses, due to excessive metal in the hub, the latter is split and 
the halves are then utilized for clamping the gear to the shaft. 
Such a gear is shown in Fig. 178 of Chapter XIV. Frequently 
it is desirable to use a split construction, by which is meant the 
gear is made in two halves that are bolted together. This is a 
common form of construction in railway motor gears, and as 
usually made, the joint comes along an arm. 

In Fig. 144 is shown the design of a triple-staggered-tooth spur 
gear designed and constructed by the Mesta Machine Co. of 
Pittsburg. The design is quite different from that ordinarily 
used, in that the face of the gear is built up of three separate rims 
bolted together with the teeth in the three sections arranged in a 



Art. 229] 



LARGE GEARS 



307 



staggered order. The gear is actually constructed of six sepa- 
rate parts. The central part or spider is split through the hub 
and rim between two arms and has bolted to it the separate 
outside rims, each of which consists of two halves. The gear is 
used for driving a sheet mill and is capable of transmitting 1,600 
horse power at a pitch line speed of 2,000 feet per minute. The 
gear contains 154 teeth of 5^-inch circular pitch and has a face 
of 38 inches, and the pinion driving the gear has 20 teeth. 
Because of the high pitch line speed the drive is arranged to run 
in an oil bath. 




Another interesting design of a large gear is that in which a 
separate spider, consisting of hub and arms, has bolted to it a 
rim built up in sections. For an illustration of this type, as well 
as other designs of large gears, consult ChapterXIV. 

229. Gear-wheel Proportions. — (a) Arms. — An exact analysis 
of the stresses produced in gear arms is exceedingly difficult, and 
as far as the author is aware, no such analysis has ever been pre- 
sented. In arriving at a formula by means of which the dimen- 
sions of the arm may be calculated, we shall assume that the rim 



308 



GEAR PROPORTIONS 



[Chap. XII 



is of sufficient thickness that the load on the teeth is distrib- 
uted equally among the arms. No doubt this assumption is 
justifiable, since the rim must be made so rigid that it is subjected 
to no bending between the arms. 





(b) 



With this assumption, we get the following expression for the 
section modulus of the arm at the center of the hub : 



I = WR 

c nS 



(331) 



^~9 



{*&$& 



(a) 



t 




r~ 


\ 

mm. 









1 
\ 



^99 




m 



(b) 



Fig. 146. 



in which W is the load on the gear; R the length of the arm or, in 
this case, the radius of the gear in inches; S the allowable fiber 
stress in the material, and n the number of arms in the gear. 

By means of (331), the value of the section modulus - may be 



Art. 229] 



GEAR PROPORTIONS 



309 



determined, from which the dimensions of the adopted arm sec- 
tion may be obtained. In Figs. 145 and 146 are shown four types 
of arms that are used in gear construction. Of these, the de- 
signs shown by Fig. 145(6) and Fig. 146(a) and (o) are used 
chiefly for large and heavy gears, while the elliptical arm shown 
in Fig. 145(a) is intended for lighter service, although very often 
it is also used for heavy work. The proportions of the various 
arm sections illustrated in the above figures are given in Fig. 147. 
The dimensions of the arm at the pitch line are generally made 
approximately seven-tenths of those at the center. Since the 
elliptical cross-section is used largely for the ordinary gears, we 
shall derive a formula for that section assuming the proportions 




given in Fig. 147(a). The section modulus for an ellipse, having 
the proportions referred to above, is -wr; hence from (331), we 
get 

(332) 



-4 



2 WR 

nS 



The number of arms in gears varies with the diameter, and the 
following represents the prevailing practice: 

1. Four or five arms for gears up to 16 or 20 inches in diameter. 

2. Six arms for gears from 16 to 60 inches in diameter. 

3. Eight arms for gears from 60 to 96 inches in diameter. 

4. Ten or twelve arms for gears above 96 inches in diameter. 
Web centers are used for smaller gears, and the thickness of the 

web approximates one-half of the circular pitch. Sometimes 
stiffening ribs are introduced between the hub and rim, and the 
thickness of such ribs is generally equal to the web thickness. 

(&) Rim. — Calculations for the rim dimensions are of little 
value, and in actual designing empirical formulas are resorted to. 



310 



GEAR PROPORTIONS 



[Chap. XII 



As shown in Figs. 145 and 146, the minimum thickness of the rim 
under the teeth is made about one-half the circular pitch and 
should taper to a slightly greater thickness where the arms join 
the rim. Good design dictates that the rim should be supplied 
with a central rib or bead, as illustrated in Figs. 145 and 146. 

(c) Face of gears. — The width of the face of a gear depends in 
general upon the type of gear, whether it has cast or cut teeth, 
the class of service, and the location of the gear. If the gear is 
located between rigid bearings, the face may be made wider than 
when the gear is a considerable distance from the bearings, since 
in the latter case the deflection of the shaft due to the load on the 
gear might seriously affect the distribution of the load across a 
wide face. For cast teeth it is good practice to make the face 
from two to three times the circular pitch, while for cut teeth 
the face is made from two and one-half to six times the circular 
pitch, three to four being a fairly good average. 

(d) Hubs. — The hubs of gears are made either solid or split, as 
stated in the preceding article. In either type of hub good design 
calls for a reinforcement of metal over the key, and this 
condition is met if the hubs are proportioned according to 
suggestions offered in Figs. 145 and 146. The object of a split 
hub is to reduce the cooling stresses in the gear and at the 
same time permit any desired adjustment of the gear on the 
shaft. Keys should always be placed under an arm in the case of 
a solid hub, and in a split hub approximately at right angles to 
the center split or hub joint. The diameters and lengths of the 
hub may be made in accordance with the formulas given in Table 
74, in which d denotes the bore of the hub. These formulas 
were published by Herman Johnson in the American Machinist 
of Jan. 14, 1904, and represent the actual practice of four large 
manufacturers. 



Table 74. — Dimensions op Gear Hubs 





Diameter 




Type of service 


Cast iron 


Steel casting 


Length 


Heavy load and great shock 
Medium load and medium 


2d. 
1.75 d +0.125" 

1.625 d+0.125" 


1.75d+0.125" 
1.625 d+0.1875" 

1.5d+0.25 


1.75dto2>id 


Light load and no shock 





Art. 230] 



METHODS OF STRENGTHENING 



311 



230. Methods of Strengthening Gear Teeth. — Occasionally it 
is desirable to have the teeth of a gear extra strong, and to obtain 
additional strength any one of the seven following methods may 
be used: (a) shrouding; (6) use of short teeth; (c) increase of the 
angle of obliquity; (d) use of stub teeth; (e) use of unequal adden- 
dum teeth; (/) use of buttressed teeth; (g) use of helical teeth. 

(a) Shrouding. — The gain in strength due to shrouding depends 
upon the face of the gear, the effect being more marked in the 
case of a narrow face than in a wider one. Wilfred Lewis con- 
siders shrouding bad practice. However, when the face is two 
and one-half times the circular pitch, he has demonstrated by 
an approximate theoretical investigation that single shrouding 
similar to that illustrated by Fig. 148(a) will increase the 
strength of the tooth at least 10 per cent, and double shrouding 




![-'♦ 




K, 


i 


'WM 


|p 



(b) 
Fig. 148. 



<r i 




p 

3 

n 


i 




WM 


i 


W//, 



(c) 



as shown in Fig. 148(6), at least 30 per cent. In many cases 
shrouding of gears is necessary, and the proportions given in Fig. 
148 for the three methods of shrouding will serve as a guide. 

(6) Short teeth. — Gear teeth whose heights are less than those 
given by common proportions are considerably stronger, and 
furthermore, they run with less noise. In America, C. W. Hunt 
advocates this type of tooth, and the following proportions 
for cast involute teeth are those he has successfully used on 
gears for coal-hoisting engines and similar machinery. 



Addendum = 0.2 p f 
Face of gear = 2 p' + 1" 
Clearance = 0.05 (p' + 1' 



(333) 



In Table 75 are given the working, as well as maximum, loads 
recommended by Mr. Hunt for a cast-iron spur gear having 20 
teeth, which is the smallest gear he uses. For proportions of 
short cut teeth recommended by Mr. Hunt, see Table 71. 



312 METHODS OF STRENGTHENING [Chap. XII 

Table 75. — Strength of Gear Teeth Used by C. W. Hunt 





Load in pounds 


Circular 
pitch 


Load in pounds 


pitch 


Working 


Maximum 


Working 


Maximum 


1 

m 
m 


1,320 
2,300 
3,000 
4,100 


1,650 
2,600 
3,700 
5,000 


2H 

3 


6,700 
8,300 
10,000 
12,000 


8,300 
10,500 
12,500 
14,800 



(c) Increase of the angle of obliquity. — The gain in strength due 
to an increase of the angle of obliquity is shown in Fig. 149. 
This figure shows the left half of a tooth having a 22^-degree 
angle of obliquity and the right half of a tooth having the same 
pitch but with an angle of obliquity equal to 15 degrees. Now 

2x 
the factor y which appears in the Lewis formula is equal to o — ,' 

from which it is evident that an increase of x, when the circu- 
lar pitch p' remains constant, 
will result in an increase of y 
5 " and consequently an increase 
in the strength of the tooth. 
This increase of x is shown 
in the figure. . 

A further advantage aside 
from the increase of strength 
lies in the fact that the size 
of the smallest pinion which 
will mesh with a rack without 
correction for interference di- 
minishes rapidly as the angle 
of obliquity increases. Thus 
with an angle of obliquity of 15 degrees, the 30-tooth pinion is 
the smallest one that can be used without correction, while with 
an obliquity of 22^ degrees, the smallest gear in an uncorrected 
set has theoretically 14 teeth, but practically this number may 
be reduced to 12. 

(d) Stub teeth. — Another method of strengthening gear teeth, 
which is now being used extensively in automobile transmission 
gears and in gears used in machine tools and hoisting machinery 
consists of a combination of (b) and (c). This combination gives 
what is known as the stub tooth. There are two systems of stub 




Fig. 149. 



Aet. 230] 



METHODS OF STRENGTHENING 



313 



Table 76.— Dimensions of the 
Fellows Stub Teeth 



teeth, differing in the detail dimensions of the teeth, as shown 
below, but agreeing on the choice of the angle of obliquity, 
namely 20 degrees. 

In one of these systems, originated by Mr. C. H. Logue, the 
proportions given in Table 71 are 
used. 

To the second system, that 
recommended by the Fellows Gear 
Shaper Co., the tooth dimensions 
listed in Table 76 apply. 

(e) Unequal addendum gears. — 
Both in Europe and in the United 
States certain manufacturers have 
advocated the use of a system of 
gearing in which the addendum of 
the driving pinion is made long, 
while that of the driven gear is 
made short, as shown in Fig. 150. 
Some of the advantages claimed for this system of gearing, after 
several years' of actual experience with it, are the following : 

1. This form of tooth obviates interference, thus doing away 
with undercut on the gears having the smaller numbers of teeth, 
and at the same time it increases the strength of such gears. 



Pitch 


Thick- 
ness on 
the pitch 
line 


Adden- 
dum 


Deden- 
dum 


% 


0.3925 


0.2000 


0.2500 


K 


0.3180 


0.1429 


0.1785 


% 


0.2617 


0.1250 


0.1562 


"A 


0.2243 


0.1110 


0.1389 


Ho 


0.1962 


0.1000 


0.1250 


%i 


0.1744 


0.0909 


0.1137 


X K 2 


0.1570 


0.0833 


0.1042 


% 


0.1308 


0.0714 


0.0893 





(a) 



(b) 



Fig. 150. 



2. The sliding friction between the flanks of the teeth is de- 
creased, since the arc of approach is shortened; hence the wear of 
the teeth is diminished. 

3. High-speed gears equipped with unequal addendum teeth 
run more quietly than standard addendum gears. 

4. With this system of gearing it is possible to make the teeth 



314 



METHODS OF STRENGTHENING 



[Chap. XII 



of the pinion and gear of equal strength, while with the standard 
system this is impossible without resorting to the use of different 
materials for the pinion and the gear. 

The tooth profile of a 15-tooth pinion having a tooth of stand- 
ard proportions is shown in Fig. 150(a), while Fig. 150(6) illus- 
trates the tooth outline of a pinion having the same number of 
teeth and the same pressure angle, but with the addendum and 
dedendum based on the Gleason standard given in a following 
paragraph. An inspection of these profiles shows clearly how the 
teeth of a pinion are strengthened by means of this system of gear 
teeth. 

From the above discussion it is apparent that the use of unequal 
addendums is desirable for gears having 
a high velocity ratio. At the present 
time unequal addendum gears are used 
extensively on the rear axle drive of 
automobiles, as quite a number of manu- 
facturers have now adopted this system 
of teeth for their bevel gears. In 
America up to the present time, the un- 
equal addendum teeth are used chiefly 
with bevel gears, but there is no reason 
why they should not be used to advan- 
tage in certain spur gear drives on ma- 
chine tools and in other classes of ma- 
chinery. As yet very little progress has been made in this 
direction. 

Gleason standard. — The Gleason Works have adopted as their 
standard for high ratio bevel gears the following proportions for 
unequal addendum teeth. 



Table 77. — Constants 
for Determining Tooth 
Thickness for Gleason 
Unequal Addendum 
Teeth 





Constant for 


tooth 
thrust • 


Pinion 


Gear 


14^° 

15° 

20° 


0.5659 
0.5683 
0.5927 


0.4341 
0.4317 
0.4073 



Addendum for pinion = 0.7 working depth 
Addendum for gear = 0.3 working depth 



(334) 



The working depth is assumed to be twice the reciprocal of the 
diametral pitch, or the circular pitch multiplied by the factor 
0.3183. 

To determine the thickness of the tooth on the pitch circle, 
when these formulas are used, multiply the circular pitch by the 
constants given in Table 77. 

(/) Buttressed tooth. — The buttress or hook-tooth gear can be 
used in cases where the power is always transmitted in the same 



Art. 231] 



SPECIAL GEARS 



315 



direction. The load side of the tooth has the usual standard 
profile, while the back side of the tooth has a greater angle of 
obliquity as shown in Fig. 151. To compare its strength with 
that of the standard tooth, use the following method: Make a 
drawing of the two teeth and measure their thicknesses at the 
tops of the fillets; then the strength of the hook tooth is to the 
standard as the square of the tooth thickness is to the square of 
the thickness of the standard tooth. 

(g) Helical teeth. — Properly supported gears having accurately 
made helical teeth will run much smoother than ordinary spur 
gears. In the latter form of gearing there is a time in each period 
of contact when the load is concentrated on the upper edge of 
the tooth, thus having a leverage equal to the height of the tooth. 




Fig. 151. 

With helical gearing, however, the points of contact at any in- 
stant are distributed over the entire working surface of the tooth 
or such parts of two teeth in contact at the same time. There- 
fore, the mean lever arm with which the load may act in order to 
break the tooth cannot be more than half the height of the tooth. 
It follows that the helical teeth are considerably stronger than the 
straight ones. The subject of helical gears will be discussed more 
in detail in Chapter XIV. 

231. Special Gears. — Specially designed gears, differing radi- 
cally from those discussed in the preceding articles, are used when 
it is desired to provide some slippage so as to protect a motor 
against excessive overload, or to prevent breakage of some part 
of the machine. Special gears are also required where heavy 
shocks must be absorbed, thus again protecting the machine 
against possible damage. The first type of gear mentioned is 



316 



SLIP GEARS 



[Chap. XII 



known as a slip gear and the second, as a flexible or spring cush- 
ioned gear. 

(a) Slip gears. — A slip gear is a combination of a gear and, a 
friction clutch, the latter being so arranged that it is always in 
engagement, but will slip when an extra heavy load comes upon 
the gear. Slip gears are used to some extent in connection with 
electric motor drives, and in such installations they really serve 




as safety devices by protecting the motor from dangerous over- 
loads. Two rather simple designs of slip gears are illustrated in 
Figs. 152 and 153. 

1. Pawlings-Harnischfeger type. — The design shown in Fig. 152 
is used by the Pawlings-Harnischfeger Co. in connection with 
some of their motor-driven jib cranes. The gear a, meshing 
directly with the motor pinion, is mounted upon the flanged hub 
b and the bronze cone c, both of which are keyed to the driven 
shaft d as shown in the figure. By means of the three tempered 



Art. 231] 



SLIP GEARS 



317 



steel spring washers e and the two adjusting nuts /, the desired 
axial force may be placed on the clutch members b and c. In 
reality the combination a, b, and c is nothing more than a com- 
bined cone and disc clutch, the analysis of which is given in de- 
tail in Chapter XVI. The angle that an element of the cone 
makes with the axis is 15 degrees for the design shown in Fig. 152. 
2. Ingersoll type. — A second design of slip gear differing slightly 
from the above is shown in Fig. 153. It is used by the Ingersoll 
Milling Machine Co. on the table feed mechanism of their 
heavy milling machines. Its function is to permit the pinion d 
to slip when the load on the cutter becomes excessive. The 




Fig. 153. 



driving shaft a has keyed to it a bronze sleeve b upon which slides 
the sleeve c, also made of bronze. As shown, a part of the length 
of the sleeves b and c is turned conical so as to fit the conical bore 
of the steel pinion d. The frictional force necessary to operate 
the table is obtained by virtue of the pressure of the spring e 
located on the inside of the sleeve c. By means of the adjusting 
nuts / and g, the spring pressure may be varied to suit any condi- 
tion of operation. 

(b) Flexible gears. — The so-called flexible or spring-cushioned 
gear is used on heavy electric locomotives, and its chief function 
is to relieve the motor and entire equipment from the enormous 
shocks due to suddenly applied loads. In Fig. 154 is shown a 
well-designed gear of this kind as made by The R. D. Nuttal] Co. 
The gear consists of a forged-steel rim a on the inner surface of 
which are a number of short arms or lugs b as illustrated in Fig. 



318 



FLEXIBLE GEARS 



[Chap. XII 



154(6), which represents a section through the gear along the line 
OB. The gear rim a is mounted upon the steel casting hub c 
which is equipped with projecting arms d. These arms are 
double, as shown in the section through OB, and are provided 
with sufficient clearance to accommodate the projecting lugs b. 
The heavy springs e form the only connection between the double 
arms d and the lugs b; hence, all of the power transmitted from 
the rim to the hub must pass through the springs e. Special 
trunnions or end pieces are used on the springs to give a proper 
bearing on the lugs and arms. The cover plate / bolted to the 
hub c affords a protection to the interior of the gear against 
dust and grit. It is evident from this description that the springs 




Fig. 154. 

provide the necessary cushioning effect required to absorb the 
shocks caused by suddenly applied overloads. The remarks 
relating to the design of spring-cushioned sprockets, as given in 
Art. 193, also apply in a general way to the design of flexible 
gears. 

EFFICIENCY OF SPUR GEARING 



There is probably no method of transmitting power between 
two parallel shafts that shows a better efficiency than a pair of 
well-designed and accurately cut gears. So far as the author 
knows, no extensive investigation has ever been made of the effi- 
ciency of spur, bevel, and helical gearing ; at least, very little in- 
formation has appeared in the technical press on this important 
subject. It is generally assumed that the efficiency of gearing 
becomes less as the gear ratio increases, and the correctness of 
this assumption is proved by a mathematical analysis proposed 
by Weisbach. 



Art. 232] 



EFFICIENCY OF GEARS 



319 



232. Efficiency of Spur Gears. — By means of the analysis 
following, it is possible to arrive at the expression for the amount 
of work lost due to the friction between the teeth. Knowing 
this lost work, also the useful work transmitted by the gears, we 
have a means of arriving at the probable efficiency of a pair of 
gears. 

In Fig. 155 are shown two spur gears 1 and 2 transmitting 
power. In this figure the line MN, making an angle /3 with the 
common tangent CT, represents the line of action of the tooth 
thrust between the gears. 




Fig. 155. 



Let Wi and n 2 denote the revolutions per second of the gears. 
T\ and T 2 denote the number of teeth in the gears 1 and 2, 

respectively. 
o} X and co 2 denote the angular velocity of the gears 1 and 2, 
respectively. 

p' = the circular pitch. 
s = the distance from the pitch point C to the point 

of contact of two teeth, 
ju = coefficient of sliding friction. 

To find the velocity of sliding at the point of contact of two 
teeth, we employ the principle that the relative angular velocity 
of the gears 1 and 2 is equal to the sum or difference of the angular 
velocities a>i and co 2 of the wheels relative to their fixed centers 
Oi and 0z\ thus if co denotes this relative angular velocity, 



320 EFFICIENCY OF GEARS [Chap. XII 

the minus sign being used when one of the wheels is annular and 
coi and co 2 have the same sense. The velocity v' with which one 
tooth slides on the other is then the product of this angular veloc- 
ity o) and the distance s between the point of contact and the 
pitch point, which is the instantaneous center of the relative 
motion of 1 and 2; that is, 

v' = s (coi ± w 2 ) (335) 

The distance s varies ; at the pitch point it is zero, and when the 
teeth quit contact it has a value of 0.7 to 0.9 p' with teeth having 
the ordinary proportions. The average value of s may be taken 
as 0.4 p f . 

Since P is the normal pressure between the tooth surfaces, the 
force of friction is imP, and the work of friction per second is 

W t = vPv' (336) 

The formula for W t may be put into more convenient form by 
combining (335) and (336), and substituting in the resulting equa- 
tion the following values of «i, co 2 and n 2 : 

T 

wi = 2tUi; co 2 = 2 7rn 2 ; n 2 = n\ 7=- 

I 2 

Hence, 

W t = 0.8nirPv[±±^, (337) 

in which v represents the velocity of a point on the pitch line. 

The component of P, in the direction of the common tangent 
CT to the pitch circles of the gears, is Pcos/3; hence the work per 
second that this force can do is 

Wo = PvcosP (338) 

Adding (337) and (338), it is evident that the work W put into 
the gears, omitting the friction on the gear shafts, is 

W' = Wo+W t (339) 

The component of P in a radial direction is Psin/3, and repre- 
sents the force that causes a pressure upon the shaft bearings. 
The work lost in overcoming the frictional resistances of these 
bearings is as follows: 

W h = irAiTsinj8(nidi + n 2 d 2 ), (340) 

in which d x and d% represent the diameters of the shafts, and n' 
the coefficient of journal friction. 



Art. 232] EFFICIENCY OF GEARS 321 

With friction considered, it follows that the total work required 
to transmit the useful work Wo is 

W = Wo + W t + W b (341) 

The efficiency of the pair of gears including the bearings is 
therefore 

i = w (342) 

If it is desirable to estimate the efficiency of the gears exclusive 
of the bearings, the following expression may be used : 

, Wo 1 m ,> 

71 = W = ~ ~* — ^ 



1 + 2.51 M sec |8 [± ± ~J 



For gears having cast teeth, the coefficient of friction n may 
vary from 0.10 to 0.20, while for cut gears the value may be less 
than one-half those just given. However, even with the larger 
coefficient of friction quoted, the loss due to friction is small. 
It is apparent from (343) that the efficiency is increased by em- 
ploying gears having relatively large numbers of teeth. 

References 

American Machinist Gear Book, by C. H. Logtje. 

A Treatise on Gear Wheels, by G. B. Grant. 

Machine Design, by Smith and Marx. 

Spur and Bevel Gearing, by Machinery. 

Elements of Machine Design, by J. F. Klein. 

Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 

Elektrisher Antriebmittels Zahnradubertragung, Zeit. des Ver. deutsch 
Ing., p. 1417, 1899. 

Interchangeable Involute Gear Tooth System, A. S. M. E., vol. 30, p. 921. 

Interchangeable Involute Gearing, A. S. M. E., vol. 32, p. 823. 

Proposed Standard Systems of Gear Teeth, Amer. Mach., Feb. 25, 1909. 

Tooth Gearing, A. S. M. E., vol. 32, p. 807. 

Gears for Machine Tool Drives, A. S. M. E., vol. 35, p. 785. 

The Strength of Gear Teeth, A. S. M. E., vol. 34, p. 1323. 

The Strength of Gear Teeth, A. S. M. E., vol. 37, p. 503. 

Recent Developments in the Heat Treatment of Railway Gearing, Proc. 
The Engrs. Soc. of W. Pa., vol. 30, p. 737. 

Gear Teeth Without Interference or Undercutting, Mchy., vol. 22, p. 391. 

Spur Gearing, Trans. Inst, of Mech. Engr., May, 1916. 

Safe and Noiseless Operation of Cut Gears, Amer. Mach., vol. 45, p. 1029. 

Efficiency of Gears, Amer. Mach., Jan. 12, 1905; Aug. 19, 1909. 

Internal Spur Gearing, Mchy., vol. 23, p. 405. 

Chart for Selecting Rawhide Pinions, Mchy., vol. 23, p. 223. 



CHAPTER XIII 
BEVEL GEARING 

When two shafts which intersect each other are to be connected 
by gearing, the result is a pair of bevel gears. Occasionally, 
however, the shafts are inclined at an angle to each other but do 
not intersect, in which case the gears are called skew bevels. 
The form of tooth which is almost universally used for bevel 
gears is the well-known involute. This is probably due to the 
fact that slight errors in its form are not nearly so detrimental to 
the proper running of the gears as when the tooth curves are 
cycloidal. 

233. Methods of Manufacture. — Bevel gears may be either 
cast or cut. The process of casting is not materially different 
from that used in spur gearing, but the process of cutting is much 
more difficult on account of the continuously changing form and 
size of the tooth from one end to the other. 

As in the case of spur gearing, there are several different 
methods of cutting the teeth, some of which form the teeth with 
theoretical accuracy, while others produce only approximately 
correct forms. Three of the methods give very accurate results, 
but they require expensive special machines and are used only 
when very high-grade work is desired. The three methods are: 
the templet-planing process, represented by the Gleason gear 
planer; the templet-grinding process, now used but little, repre- 
sented by a machine manufactured by the Leland and Faulconer 
Co.; and the moulding-planing process, represented by the Bil- 
gram bevel gear planer. 

In each of these processes the path of the cutting tool passes 
through the apex of the cone, that is, the point of intersection of 
the two shafts, and consequently the proper convergence is 
given to the tooth. With a formed rotating cutter, it is impossi- 
ble to produce the proper convergence and in many cases the 
teeth have to be filed after they are cut, before they will mesh 
properly. Nevertheless, the milling machine is very commonly 
used for cutting bevel gears, for the simple reason that the 

322 



Art. 234] 



BEVEL GEAR TEETH 



323 



equipment of most shops includes a milling machine, while 
comparatively few shops do enough bevel gear cutting to justify 
the purchase of an expensive special machine for that purpose. 

234. Form of Teeth. — When the gears are plain bevel frictions, 
it is evident that the faces of the gears must be frustums of a 
pair of cones whose vertices are at the point of intersection of the 
axes. These cones may now be considered the pitch cones of a 
pair of tooth gears, and the teeth may be generated in a manner 
analagous to the methods used for spur gearing. In discussing 




the method of forming the teeth, the involute system only will be 
considered, since the cycloidal forms are seldom used. 

In Fig. 156, let the cone OH I represent the so-called base cone 
of the bevel gear shown, from which the involute tooth surfaces 
are to be developed. In order to simplify the conception of the 
process of developing, imagine the base cone to be enclosed in a 
very thin flexible covering which is cut along the line OE. Now 
unwrap the covering, taking care to keep it perfectly tight; then 
the surface generated by the edge or element OF is the desired 
involute surface. The point E, while it evidently generates an 
involute of the circle HI, is also constrained to remain a constant 



324 



BEVEL GEAR TEETH 



[Chap. XIII 



distance from equal to OE, or in other words, it travels on the 
surface of a sphere HAI. For that reason the curve EF is called 
a spherical involute. The spherical surface which should theo- 
retically form the tooth profile is a difficult surface to deal with in 
practice on account of its undevelopable character, and as is 
shown in the figure no appreciable error is introduced if the con- 
ical surface CBD is substituted for the spherical surface CAD. 
The cone CBD, which is called the back cone, is tangent to the 




sphere at the circle CD, and the pitch distance practically coin- 
cides with the sphere for the short distance necessary to include 
the entire tooth profile. When it is desired to obtain the form of 
the teeth, as is necessary in case a wood pattern or a formed cutter 
is to be made, the back cone is developed on a plane surface as 
shown in Fig. 157. It is evident that the surface which contains 
the tooth profile has a radius of curvature equal to BD, so the 
profile must be laid off on a circle of that radius in precisely the 
same manner as that used for spur gearing. However, this pro- 
file is correct for one point only, namely, at the large end. In 



Art. 235] DEFINITIONS 325 

order to determine the form of the tooth for its entire length, it is 
necessary to have the profile of the tooth at each end. This 
may be obtained by developing the back cone AOG and proceed- 
ing as before. The two profiles just discussed are laid out from 
the line LI as shown. The back cone radius LK is equal in length 
to AO, and LJ is equal to BD. If a wood pattern is to be made, 
templets are formed of the exact profile of the tooth at the large 
and small ends. These templets are then wrapped around the 
gear blank and the material is cut out to the shape of the templets. 

235. Definitions. — (a) By the expression back cone radius is 
meant the length of an element of the back cone, as for example 
the line IJ in Fig. 157. 

(6) The edge angle is the angle between a plane which is tangent 
to the back cone and the plane containing the pitch circle. In 
Fig. 157 this angle is designated by the symbols 0i and 2 for 
the pinion and gear, respectively. 

(c) The center angle is the angle between a plane tangent to the 
pitch cone and the axis of the gear. For the pinion and gear 
shown in Fig. 157, the center angle is designated as a.\ and a 2 , 
respectively. From the geometry of the figure it is evident that 
0i = a i, and 2 = «2. 

(d) The cutting angle, represented by the symbols Xi and X 2 in 
Fig. 157, is the angle between a plane tangent to the root cone and 
the axis of the gear. 

(e) By the term face angle is meant the angle between the plane 
containing the pitch circle and the outside edge of the tooth, as 
represented by the Symbols <pi and <p 2 in Fig. 157. 

(/) Backing is the distance from the addendum at the large 
end of the teeth to the end of the hub, as represented by the 
dimensions L x and L 2 in Fig. 157. 

(g) The expression formative number of teeth is the number of 
teeth of the given pitch which would be contained in a complete 
spur gear having a radius equal to the back cone radius. This 
number of teeth is used in selecting the proper cutter for cutting 
the gear and also for obtaining the value of the Lewis factor when 
calculating the strength of the bevel gear. 

BEVEL-GEAR FORMULAS 

The following formulas, expressing the relations existing be- 
tween the various dimensions and angles of bevel gears, are im- 



326 



ACUTE-ANGLE BEVEL GEARS 



[Chap. XIII 



portant and are necessary for determining the complete dimen- 
sions required to manufacture such gears. In arriving at these 
formulas, two general types of bevel gears must be considered : 

1. That type in which the angle between the intersecting shafts 
is less than 90 degrees, as shown in Fig. 158. 

2. That type having an angle between the shafts greater than 
90 degrees, an illustration of which is shown in Fig. 159. 

Having obtained the formulas for either of these types of gears, 
those for the more common case, namely when the shafts make an 
angle of 90 degrees, may readily be derived. 

236. Acute-angle Bevel Gears. — In Fig. 158 is shown a pair of 
bevel gears in which the angle 6 between the shafts is less than 
90 degrees. In deriving the desired relations, the following no- 
tation will be used: 




D 

D' 
T 



the pitch diameter. 

the outside diameter. 

the number of teeth. 

the diameter increment. 

the clearance at the top of the tooth. 

the diametral pitch. 

the circular pitch. 

the addendum. 



Art. 236] ACUTE-ANGLE BEVEL GEARS 327 

In this discussion the subscripts 1 and 2, when applied to the 
various symbols, refer to the pinion and gear, respectively. From 
the geometry of the figure, we obtain the following relations : 

Z>2 



£>i sin 6 







2 sin 6 




b 


2 tan 6 


tan 


a\ 


2 (a + b) 
sin 6 



D 2 + D t cos 6 



(344) 

-J + cos 6 
-* i 

The equation just established enables us to determine the mag- 
nitude of the center angle of the pinion. Subtracting a-i from the 
angle 9 included between the two shafts gives the magnitude of 
the center angle a 2 of the gear. 

If it is desired to determine the angle a 2 by means of calcula- 
tions, the following formula, derived in the same manner as (344), 
may be used : 

tan a 2 = m (345) 

Y 2 + cos ° 

Determining the magnitudes of «i and a 2 by means of (344) and 
(345), the calculations may be checked very readily, since 
ol\. + a 2 = d. 

To determine the face angle $i of the pinion, we must find the 
angle increment, by which is meant the angle included between 
the pitch cone element and the face of the tooth. Thus 

2 s 

tan (0i — ai ) = — sin a h (346) 

D\ 

from which the angle increment may be obtained. The addition 
of (|3i — ai) to the center angle gives the magnitude of the face 
angle. 

The angle decrement (cm — \i) may be determined from the 
following relation: 

tan (ax - XO = 2(6 p +c) sin «i (347) 

By subtracting (an — X x ) from the center angle the magnitude 
of the cutting angle \ x is found. 



328 OBTUSE-ANGLE BEVEL GEARS [Chap. XIII 

Since the angle increment and angle decrement of the pinion 
are exactly the same as the corresponding angles of the gear, the 
face and cutting angles of the latter may be found. 

In turning the blanks, it is necessary that the outside diameter 
of both the pinion and the gear be known. These diameters are 
obtained by adding twice the diameter increment to the pitch 
diameters. The diameter increment is calculated by the follow- 
ing equations : 

For the pinion, e x = scosail (oaq\ 

For the gear, e 2 = scosa 2 j 



From these relations we get 

. -J- O. O fflS/v, 

(349) 



D x = Di + 2scosq;i 
D 2 = D 2 + 2 s cosa 2 



The length of the face of the pinion measured parallel to the 
axis is i^cos/Si and the corresponding dimension for the gear is 
Fcos/3 2 . 

237. Obtuse-angle Bevel Gears. — By the expression obtuse- 
angle bevel gearing is meant a gear and pinion in which the angle 
between the shafts is more than 90 degrees. It is evident from 
this that the following three forms of such gearing are possible: 

(a) In the first form, which is more common than either of the 
other two, the center angle a 2 of the gear is made less than 90 
degrees. For convenience of reference, we shall call this form 
the regular obtuse-angle bevel gear. 

(b) In the second form, which is rarely used, the center angle 
a 2 of the gear is made 90 degrees. In this case the pitch cone 
becomes a plain disc; such a gear is then known as a crown gear. 

(c) In the third form, which should be avoided whenever pos- 
sible, the center angle a 2 of the gear is greater than 90 degrees. 
In such a gear the teeth must be formed on the internal conical 
surface, thus giving it the name of internal bevel gear. An internal 
bevel gear can generally be avoided without changing the posi- 
tions of the shafts by using an acute-angle gear set, in which the 
angle between the shafts is made equal to the supplement of the 
original angle between the shafts. 

Using the same notation as in the preceding article, the. im- 
portant formulas for the bevel gears illustrated in Fig. 159, in 
which the angle 6 is greater than 90 degrees, are as follows : 



Art. 237] 



OBTUSE-ANGLE BEVEL GEARS 



329 



For the pinion 



tan a i 



sin (180 - d) 






cos (180 



sin 6 



T 1 



+ cos 



(350) 



Generally speaking, the first form of equation (350) is preferred 
by most designers and shop men, although the second form, which 
is the same as (344), is really more convenient. In the solution 
of any problem pertaining to obtuse-angle bevel gearing, it is 
well to determine what form of obtuse bevel gear is being ob- 




tained before proceeding with the calculations, as forms (b) and 
(c) discussed above require special formulas. To find out what 
form of gear is being obtained proceed in the following manner: 

To the magnitude of a h obtained from (350), add 90 degrees 
and if the sum thus obtained is in excess of the given angle 0, 
then the resulting gears will be of form (a), namely ordinary ob- 
tuse bevel gears. If, however, the sum a x -f 90 is equal to the 
given angle 9, the result will be a crown gear and pinion. An 
internal bevel gear will result when («i + 90) <6. 

For the ordinary obtuse-angle bevel gear, the center angle a 2 



330 STRENGTH OF BEVEL GEARS [Chap. XIII 

of the gear, if desired, may be determined by means of the fol- 
lowing formula : 

sin (180 - 0) sin0 /OPi . 

tana 2 = ^ — = jj (351) 

~ - cos (180 - 0) ~ + cos 

i 2 1 2 

The remaining calculations for the ordinary obtuse-angle gears 
are made by means of the formulas given in the preceding article. 

238. Right-angle Bevel Gears. — The great majority of the 
bevel gears in common use in machine construction have their 
shafts at right angles to each other as shown in Fig. 157. The 
formulas in this case may be derived directly from those in Art. 
236, by substituting for its magnitude 90 degrees; hence (344) 
and (345) reduce to the following simple forms: 



tan «! = — 

1 2 

tan a 2 = jfr 
J- 1 



(352) 



The remaining formulas given in Art. 236 will apply to the 
present case without change or modification. 

STRENGTH OF BEVEL GEARING 

239. General Assumptions. — As in the case of spur gearing, 
formulas for the strength of bevel-gear teeth will be derived for 
the following two cases: (a) When the teeth are cast; (6) when 
the teeth are cut. In analyzing the strength of both kinds of 
teeth, we shall assume that the gear is supported rigidly and that 
the load coming upon it will not distort the teeth. Distortion 
of the teeth means that the elements of the tooth form will no 
longer intersect at the apex of the pitch cone. The above as- 
sumption also means that the distribution of the load on the tooth 
produces equal stresses at all points along the line of the weakest 
section. The last statement may be proved by the following 
analysis: From Fig. 160 or 162 it is evident that the dimensions 
of the cross-section of the tooth, at any section, are proportional 
to the distance that the section is from the apex 0; hence we ob- 
tain the following series of equations: 

hi } 



fl = -r- 



(353) 



Art. 240] 



STRENGTH OF BEVEL GEARS 



331 



Furthermore, the deflection A of the tooth at the point where the 
line of action of the force dW intersects the center line of the 
tooth is also proportional to the distance I. 

The deflection of the small section dl is given by the expression 

h 3 dW 



3 EI 



Id 



(354) 



Substituting in (354) the value of h from (353) and the value of 
I in terms of the dimensions of the section, it follows that 



klEt\ 



= Kl 




(355) 



Fig. 160. 



Applying the formula for flexure to the elementary cantilever 
beam, we obtain 

hdW = ^ (356) 

o 



Combining (353) and (356), we find 

dW = St\l 
dl ~ 6 hhi 
Comparing (355) and (357), it follows that 
Kl 
CI 



s 



CIS 



constant 



(357) 



(358) 



240. Strength of Cast Teeth. — It is sufficiently accurate to 
consider the cast bevel gear tooth as a cantilever beam, the cross- 



332 STRENGTH OF CAST TEETH [Chap. XIII 

sections of which are rectangular and converge toward the apex 
of the pitch cone. Furthermore, the load to be transmitted is 
assumed as acting tangentially at the tip of the tooth. The 
formula for the strength of cast teeth based upon the above as- 
sumption, as well as that given in the preceding article, may be 
derived as follows: 

By equating the bending moment on a small element dl of the 
tooth to its moment of resistance and solving for the elementary 
force dWi we have from (356) that 

St*dl 



Also, from (357) we get 



dw m 



dW = §$W (359) 

6lihx 



Now the moment of the elementary force dW about the apex 
is IdW; hence the elementary moment 

dM = -^-j- 

Integrating this expression between the limits h and h, we ob- 
tain 

M = Ts^T $ " ® (360) 

18 hih 

Since M represents the total turning moment about the apex 
of the pitch cone, we may readily determine the magnitude of 
the force acting at any point, as for example at the large diameter 
of the gear, by merely dividing M by the distance from that point 
to the apex. Let Wi denote the force which, if applied at the 
large end of the tooth, will produce a turning moment equal to 
M; then 

* - SW (361) 

Substituting in (361) the value of h = h — f, and simplifying 
the resulting equation, 

The proportions of cast bevel gear teeth are the same as those 
given for cast spur gears in Art. 222, namely hi = 0.7 p' and 



Art. 240] STRENGTH OF CAST TEETH 333 

ti = 0.475 p' . Substituting these values in (362), we get 

3/ , P 



Wx = 0.018 5?'/ h ~-/+y 

3/ 



(363) 



r 3 f Pi 
Letting m denote the quantity 0.018 3 — : — h -% \, re- (363) 

L li i x J 

duces to the following form: 

Wi = Sp'fm (364) 

A study of the prevailing practice among manufacturers of 
cast bevel gears shows that the face of such gears is made from 



| 0.05- 

E 

S 0.04 
U- 

<+- 

<o 
o 

1 0.03- 












































k 






























































































































































































































































































































































^> 



































































































































































































































































































































































































































































































































































































































































































































































































0.1 0.Z 03 0.4 

Ratio of f to lj 
Fig. 161. 

two to three times the circular pitch, depending upon the diam- 
eters of the gear and pinion. Another rule that should be ob- 
served is that the ratio of /to Zi should not exceed one-third. For 
values of the permissible stress S for the various grades of cast 
materials, equation (330) and Table 72 should be used. To 
facilitate the use of (364), the values of the coefficient m for 
various ratios of / to l\ are put into the form of a graph, shown in 
Fig. 161. 

241. Strength of Cut Teeth. — The formula generally adopted 
by designers for calculating the strength of cut bevel teeth is the 
one proposed by Mr. Wilfred Lewis. The assumptions regarding 
the distribution of the tooth pressure made in the preceding 
article will also hold in the discussion of the cut teeth; hence, the 
equations (359) to (362) inclusive will hold in the present case. 
As in the analysis of the cut spur gears, Mr. Lewis considered the 



334 



STRENGTH OF CUT TEETH 



[Chap. XIII 



tooth as equivalent to a beam of uniform strength, that is, one 
having a parabolic cross-section as indicated in Fig. 162. From 
the geometry of the figure, it is evident that 

t\ = 4 hxXx 

and substituting this value in (362), and multiplying and dividing 
through by p' , we get 



s P 



(365) 




Fig. 162. 



Now the ratio of 2x to Sp f is simply the so-called Lewis factor 
discussed in detail in Art. 223; hence, replacing it by the symbol 

r / f 2 ~\ 
y and denoting the factor 1 — - + — j by the symbol n, 

L Li oil 

equation (365) may be written 

W 1 = Sp'fyn (366) 

As stated in the discussion of cast teeth, the ratio of / to h 
should not exceed one-third and the face of the gear is usually 
from two to three times the circular pitch. Equation (330) and 
the data contained in Table 72 should be used for arriving at 
the permissible fiber stress for the given material and given con- 
dition of operation. It is important to note that the coefficient 
n represents the ratio that the strength of the bevel gear bears to 
the strength of a spur gear of the same face and pitch. The 



Art. 241] 



STRENGTH OF CUT TEETH 



335 



graph given in Fig. 163 shows the relation existing between n and 
the ratio of / to h, and will serve as a time saver in the solution of 
bevel gear problems. 

The proportions of cut bevel teeth for the various systems in 
general use are the same as those given in Table 71 and in Art. 
230(e). 



0.9 

c 
0.8 

+■ 

c 

'<+■ 

ti- 
ll) 

o 0.7 
O 

0.6 -1 


































































K 








































































\ 




























s 






























\ 


































\ 










































































- 




















































































s 












































s 














































s 


































































































\ 




















































Is 






















































N 




















































































































s 
































































s 
































































S 




































































s 








































































V 








































































X 

































































































































































































































































































































































0.2 
Ratio o f f to I 
Fig. 163. 



242. Method of Procedure in Problems. — In order to save 
time, the following method may be used in determining the 
strength of cut bevel gear teeth. 

(a) Since y is the form factor, its magnitude cannot be based 

upon the actual number of teeth in the gear, but must be based 

upon the so-called formative number of teeth as explained in 

Art. 235. To obtain the formative number of teeth in the pinion 

shown in Fig. 164, multiply the actual number of teeth by the 

. 2h 
ratio jz-. For the gear, the formative number is equal to the 
JJ\ 

2 1 
ratio -~ multiplied by the actual number of teeth in the gear. 

From Tables 69 and 70, determine the factor y corresponding to 
the formative number. 

(6) From Fig. 163 determine the magnitude of the coefficient 
n for the assumed ratio of / to £ x . 



336 



RESULTANT TOOTH PRESSURE 



[Chap. XIII 



(c) For the given material and the speed of the gears determine 
the magnitude of the stress S. 

(d) Knowing p' and / and having established values for y, n 
and S, the load transmitted by either the gear or pinion may be 
calculated. 

243. Resultant Tooth Pressure. — The formulas derived in 
Arts. 240 and 241, instead of giving the resultant load on the gear 
tooth, merely give an equivalent load at the large end of the tooth. 
The resultant normal tooth pressure, as well as its point of appli- 
cation, must be determined before it is possible to analyze the 
bearing loads and thrusts due to the action of bevel gears. Re- 




ferring to the bevel-gear tooth shown in Fig. 162, the normal tooth 
pressure is considered as acting along the outer edge of the tooth 
as shown in the end view of the tooth. The line of action of the 
normal pressure intersects the center line of the tooth at the point 
A, at a distance hi above the weakest section of the tooth. 

To determine the magnitude W of the resultant of all the ele- 
mentary tooth pressures dW, as well as the point of application 
of this resultant, proceed as follows: 

From (357) it is evident that 



w = cs 



Jh 



ldl = Lb (ll 



© 



Taking moments of dW about the apex 0, we get 
dM = IdW = CSPdl 



whence 



M = — (Z» 
3 V1 



S) 



(367) 



(368) 



Art. 243] 



RESULTANT TOOTH PRESSURE 



337 



The distance that the point of application of W is from the 
apex is found by dividing M by W; hence 

fc.jf.arg + fafc + gi (369 ) 

W 3 L l x + l 2 J ; 

To simplify the determination of Zo, it is best to put (369) in 

terms of the radius R shown in Fig. 165. From the geometry 
of the figure, it follows that 

Ro = U sin a (370) 




Fig. 165. 

Substituting in (369) the value of U = U — f, and reducing, we 
get 



4 sin 



3£>! 
Combining (370) and (371), we have 
4 sin 2 a 



Ro 



3Di 



Zsin 2 




(371) 



D, 



in which Z denotes the factor 



(372) 



3/ P 

3 -17 4 



2 - 



/ 



338 



BEARING PRESSURES AND THRUSTS [Chap. XIII 



To facilitate the use of the formula for R , the coefficient Z was 
determined for various values of the ratio f toll. These values 
were then plotted in the form of a graph, as shown in Fig. 166. 
By means of the graph and (372), the value of R may easily be 
calculated, since the angle a is known for any particular gear. 

Now the magnitude of the resultant tooth pressure W can be 
calculated by means of (376), but since the latter is more or less 
involved a more direct method for finding W is desirable. This is 



1.9 

N 
1.8 

+■ 
c 

<L> 
O 

<u 1.7 

o 

o 

1 1.6-1 


























































































\ 




























\ 






























\ 
































^ 










































































































. 












































- 














































































































































































s 


*. 
















































\ 




















































S 






















































\ 


























































\ 




























































N 
































































































































N 


































































S 




































































\ 


\. 




















































































































































































































N 
















































































































































































































































S 






















































































































































































_ 





























































0.2 

Ratio of f to I, 
Fig. 166. 



0.4 



obtained by dividing the torsional moment T on the gear by the 
radius R ; whence 



T 

Ro 



(373) 



244. Bearing Pressures and Thrusts. — Having determined the 
resultant tooth pressure W as well as its point of application, we 
are now prepared to discuss the pressures and thrusts coming 
upon the bearings of the supporting shaft. Letting W n in Fig. 
165 represent the resultant normal tooth pressure; then resolving 
W n along the tangent to the pitch circle, we get the resultant 
tangential tooth pressure 



W = WnCOSfi 



(374) 



Art. 245] GEAR-WHEEL PROPORTIONS 339 

The component of W n at right angles to the element of the pitch 
cone, namely, that along the line AB in Fig. 165, is 

W r = W n sinp = TftanjS (375) 

The component W produces a lateral pressure upon the sup- 
porting bearings but no thrust along the shaft of the gear. The 
component W r produces both lateral pressure and end thrust, 
the magnitudes of which are given by the following expressions : 



Lateral pressure due to W r = W r cos a = W tar /3 cos « . , 
Thrust due to W r = W r sin a = W tan sin - 



:) 



To obtain the resultant lateral pressure upon the bearings, the 
two separate components must be combined, either algebraically 
or graphically, and in order to arrive at the exact distribution of 
the resultant pressure, the location of the bearings relative to the 
gear must be established. 

Graphical methods may also be employed to determine W, 
W r , and their various components, as shown in Fig. 165. 

BEVEL-GEAR CONSTRUCTION 

In general, the constructive features of bevel gears are similar 
to those used for spur gears. Small pinions are made solid as 
shown in Fig. 158, and for economy of material larger pinions 
are made with a web. Examples of the latter construction are 
shown in Figs. 158 and 159. Not infrequently the webs are pro- 
vided with holes in order to decrease the weight of such gears. 
Large bevel gears are made with arms, the design of which will be 
discussed in the following article. Bevel gears are seldom made 
in extremely large sizes, and for that reason split or built-up 
gears are used but little. 

245. Gear-wheel Proportions. — (a) Anns. — In bevel gears the 
T-arm is remarkably well adapted for resisting the stresses that 
come upon it, and for that reason is used rather extensively in 
gears of large size. In small gears, however, the greater cost of 
the arm construction more than offsets the saving of material; 
therefore for such gears the web and solid centers are in common 
use. Fig. 167 shows a bevel gear with a T-arm. 

The rib at the back of the arm is added to give lateral stiffness, 
that is, to take care of the load component W r discussed in Art. 
244. This rib adds practically nothing to the resistance of the 



340 



GEAR-WHEEL PROPORTIONS 



[Chap. XIII 



arm to bending in the plane of the wheel, and for that reason, in 
deriving the formula for the strength of the arm, the effect of the 
rib is not considered. As in the case of spur gears, the arm is 
treated as a cantilever beam under flexure, and it is assumed that 
each arm will carry its proportionate share of the load trans- 
mitted by the gear. 

Denoting the thickness and width of the arm by 6 and h, 
respectively, and equating the external moment to the resisting 
moment, we get 

' W1D1 = Sbh? 
2n 6 ' 

in which n denotes the number of arms, and W\ and Di are the 




J I 


— p' 

2 
/ O 


w/m/A 


% • 







Fig. 167. 



equivalent load and pitch diameter, respectively, at the large end 
of the tooth. 

Solving for h, we have 

The dimension b is generally made equal to about one-half of 
the circular pitch, as shown in Fig. 167. The permissible stress 
for cast iron varies from 1,500 to 3,000 depending upon the size 
of the gear. The thickness of the rib on the back of the arm 
proper is made as shown in the figure. 



Art. 247] MOUNTING BEVEL GEARS 341 

(b) Rim and hub. — For the proportions of the rim and the rein- 
forcing bead on the inside of the rim, consult Fig. 167. The hub 
is made similar to those used for spur gears, proportions of which 
are given in Table 74. 

246. Non-metallic Bevel Gears. — Frequently where noiseless 
operation is desirable bevel gears made of rawhide and Fabroil 
are used. In Fig. 141 (6) is shown the design of a rawhide gear 
that has given excellent service. The same general constructive 
feature would be used when a Fabroil filler is employed ; but in 
place of the plain rivets, the threaded type should be used, as 
recommended by the manufacturer of such gears. In general, 
the discussion of non-metallic gears given in the preceding chapter 
applies also to bevel gears. 

247. Mounting Bevel Gears. — To obtain good service from an 
installation of bevel gears, it is important that the material used 
for the pinion and gear be chosen with some care and that the 
teeth be formed and cut accurately. These two factors alone, 
however, do not necessarily make a successful drive, as poorly 
designed mountings are frequently the source of many bevel- 
gear failures. The following important points should be observed 
in designing the mountings of a bevel-gear drive: 

1. Make the bearings and their supports rigid, and so that all 
parts may be easily assembled. 

2. Make provisions for taking care of the end thrust caused by 
the component W r discussed in Art. 244. 

3. Make provisions for lubricating the bearings and if neces- 
sary the gears themselves. 

4. Provide the gears with a dustproof guard, thus protecting 
the gears and at the same time protecting the operator of the 
machine. 

5. The shafts supporting the gears should be made large, so as 
to provide the necessary rigidity. Slight deflections of bevel-gear 
shafts produce noisy gears and cause the teeth to wear rapidly. 

(a) Solid bearing. — A rigid construction used to a considerable 
extent on machine tools is the solid bearing construction, two 
designs of which are shown in Figs. 168 and 169. In both of 
these designs the end thrusts are taken care of by the use of 
bronze washers, as shown. The bearings throughout are bronze 
bushed. The type of bevel-gear drive illustrated in Fig. 169 is 
used when the pinion is splined to its shaft. In such cases the 



342 



MOUNTING BEVEL GEARS 



[Chap. XIII 



hub of the pinion is made long, so that it may serve as a bearing. 
The heavy thrust is taken care of by the self-aligning steel 
washers, between which is located one made of bronze. In place 




Fig. 168. 



of the bronze thrust washers shown in Figs. 168 and 169, ball 
thrust bearings may be used. The latter type of bearings are 
more expensive than the bronze washers, and unless they are 




designed correctly they are liable to be troublesome. Of late, the 
type of radial ball bearing that is capable of taking a certain 
amount of thrust, in addition to the transverse load, is being 



Art. 247] 



MOUNTING BEVEL GEARS 



343 



used in connection with bevel-gear drives. The conical roller 
bearing is also adapted for use with bevel-gear transmissions. 

(b) Ball bearing. — In Fig. 170 is shown a design of a bevel-gear 
drive in which ball bearings are used throughout. This form of 
drive is used on a drill press and the details were worked out by 
The New Departure Mfg. Co., makers of ball bearings. The 
double-row ball bearings take both radial loads and thrusts, 
while the single-row ball bearing having a floating outer race 
takes only a transverse load. The double-row ball bearing on 
the horizontal driving shaft is nibunted in a shell or housing 
which is adjustable, thus providing means for getting the proper 
tooth engagement between the pinion and the gear. Necessarily, 




this form of construction will call for a bearing having a floating 
outer race at the farther end of the drive shaft. 

When it is desired to support the bevel pinion between two 
bearings, the design shown in Fig. 171 will give good results. The 
drive illustrated in this figure is one that is used on the rear axle 
of an automobile. The arrangement and selection of the various 
bearings were worked out by the Gurney Ball Bearing Co. The 
duplex bearing back of the pinion, having a thrust capacity of 
one and one-half times the radial load, is mounted rigidly in an 
adjustable cage. The bearing at the other end of the pinion shaft 
is of the radial type and, as shown, is mounted so as to permit a 
movement lengthwise of the shaft. The advantage of using the 



344 MOUNTING BEVEL GEARS [Chap. XIII 

cage construction just mentioned is that the pinion with its shaft 
and bearings may be assembled on the bench as a unit. The 
bearing to the left of the bevel gear is of a type capable of taking 
a thrust equal to the transverse load. The bearing supporting 
the other end of the differential housing to which the bevel gear 
is fastened, is also of the combined radial thrust type ; but in this 
case the thrust capacity is equivalent to one-half of the radial 
load. In Fig. 171, the differential bevels and the two axles are 
not shown, in order to bring out more clearly the other important 




Fig. 171. 

details. The type of bevel gearing used in the design just dis- 
cussed is the so-called " spiral bevel" which will be discussed in 
the following article. 

SPECIAL TYPES OF BEVEL GEARS 

248. Spiral Bevel Gears. — A special type of bevel gears called 
"spiral bevels" is now used extensively for driving the rear axles 
of automobiles. No doubt within a short time manufacturers of 
machine tools and other classes of machinery will begin to use 



Art. 2491 



SPIRAL BEVEL GEARS 



345 



spiral bevels, since they possess certain advantages over the 
straight-tooth gears. The teeth of these gears are curved on the 
arc of a circle if produced by the well-known Gleason spiral 
bevel-gear generator, or they are helical if produced on a generat- 
ing-gear planer. An illustration of the former type is shown in 
Fig. 172. 

In discussing spiral bevel gears, one should be familiar with 
certain terms or expressions that are now in common use. These 
are as follows : 

(a) Angle of spiral. — By the angle of spiral is meant the angle 
that the tangent AB to the tooth at the center of the gear face 




Fig. 172. 



makes with the element OA of the pitch cone. In Fig. 172 this 
angle is designated by the symbol a. 

(b) Direction of spiral. — The direction of the spiral is desig- 
nated as right or left hand, based upon the direction of the spiral 
on the pinion ; thus, by left-hand spiral is meant left hand on the 
pinion and right hand on the gear. 

(c) Lead. — By the term lead is meant the distance that the 
spiral advances within the face of the gear, as shown in Fig. 172. 

249. Advantages and Disadvantages. — (a) Advantages. — Among 
the advantages claimed for spiral and helical bevel gears are the 
following : 



346 SPIRAL BEVEL GEARS [Chap. XIII 

1. Due to the curvature of the teeth their engagement is grad- 
ual, thus tending to eliminate noise. The best results, accord- 
ing to the Gleason Works, are obtained when the lead of the spiral 
is made equal to one and one-quarter to one and one-half times 
the pitch of the teeth. 

2. The wear on the teeth of spiral bevel gears is no more than 
on the teeth of the common type of bevel. 

3. It has been found in practice that spiral bevel pinions permit 
of greater endwise adjustment than straight-tooth bevels, with- 
out producing excessive noise or causing bearing troubles. 

4. There is practically no difference between the load-carrying 
capacity of spiral and helical bevel gears, when compared with 
those having straight teeth. 

5. Spiral and helical bevel gears are better adapted to high- 
gear ratios, 5 and 6 to 1 giving satisfactory service, while with 
straight teeth 4^ to 1 seems to be about the dividing line between 
quiet and noisy gears when run at high speeds such as are com- 
mon in automobile transmissions. 

(6) Disadvantages. — The chief disadvantages resulting from 
the use of spiral or helical bevel gearing is the provision that must 
be made to take care of the additional thrust coming upon the 
bearings. In installations where the direction of rotation is 
reversed, the end thrust on the bearing must be taken care of in 
both directions, as the analysis given in the following article will 
show. Due to the additional end thrust on the bearing, it is 
probable that the efficiency of a spiral bevel-gear drive is slightly 
less than that obtained from a common bevel-gear drive. 

250. Bearing Loads and Thrusts. — The following analysis, 
applied to the spiral bevel gear, is based on the assumption that 
this form of tooth may be treated in a manner similar to a straight 
tooth having a spiral angle equal to the spiral angle measured at 
the center of the face, as denned in Art. 248. Furthermore, the 
friction of tooth contact will not be considered. To arrive at 
expressions for the bearing loads and thrusts, proceed as follows : 

(a) Direct rotation. — The spiral bevel gear, shown in Fig. 173, 
has an angle of spiral designated by a, and an angle of obliquity 
of tooth pressure equal to (3. Resolving the resultant normal 
tooth pressure, acting at G and represented by the vector AB, 
into three components, we have: 

1. The component DF perpendicular to the plane of the paper 



Art. 250] 



SPIRAL BEVEL GEARS 



347 



and also equal to W, the tangential force acting on the gear at G, 
is given by the following expression : 

W = ABcosa cos(3 (378) 

2. The component acting along the element of the pitch cone is 
represented by EF and its magnitude is 

EF = HG = Wtsma (379) 

3. The component at right angles to the element of the pitch 




Fig. 173. 

cone is represented by the vector BC or GI, the magnitude of 
which is 

BC = GI = ABsm(3 

tan/3 



W 



(380) 



Resolving the three forces DF, HG, and GI into components 
whose lines of action are along the center line of the shaft and at 
right angles thereto we obtain for the thrust along the shaft of 
the gear 

F y = HGcos 6 + GI sin 

W 

= (sin a cos 9 + tan /3 sin 0) (381) 



348; SPIRAL BEVEL GEARS [Chap. XIII 

and for the thrust along a line at right angles to the shaft of the 
gear, or in other words along the shaft of the pinion 

F x = HG sin - GIcos 6 

W 

= (sin a sin d - tan 8 cos 0) (382) 

cos a 

It follows that the thrust exerted by the pinion upon its shaft has 
a magnitude given by (382), but its direction is opposite to that 
oiFi. 

(6) Reversed rotation. — Supposing now that the direction of 
rotation of the gear is reversed, the component along the element 
of the cone, given by (379), reverses in direction, or in other 
words, it acts toward the point in Fig. 173; thus 

EF = GH = -W tana (383) 

Furthermore, the component BC or GI at right angles to the cone 
element remains as in the preceding case. 

Resolving DF, GH, and GI, as in the preceding case, we get for 
the thrust along the shaft of the gear 

F y = GI sin + GH cos 9 

W 

= (tan 8 sin — sin a cos 0) (384) 

cos a 

In a similar manner, the magnitude of the thrust along the 
pinion shaft is found to be 

F x = HG sin - GI cos 

W 

= (sin sin a + tan 8 cos 6) (385) 

cos a v 

If the spiral of the teeth is reversed for the case just discussed, 
the equations deduced for the preceding case will hold. 

251. Experimental Results. — In order to determine the actual 
thrusts upon the bevel pinion of automobile drives, the Gleason 
Works made an extensive series of tests upon various types of 
bevel gears. The results were published in Machinery, vol. 20, 
p. 690. Table 78 gives the various dimensions and angles of the 
gears and pinions, and the average pinion thrusts per 100 pounds 
of load on the tooth. The pinion thrusts have been calculated by 
substituting in (382) and (385) the value of W and the values 
of the functions .of the various angles. Comparison of these 



Art. 252] 



SKEW BEVEL GEARS 



349 



calculated values with the actual pinion thrusts observed in the 
tests show good agreement. 

Table 78. — Experimental Data Pertaining to Bevel Gearing 









Type of bevel gearing 




Common 


Spiral tooth 


1 

2 


| Number of teeth 


f Pinion 
{ Gear 


15 
53 


14 
53 


15 
53 


3 


Pressure angle — /3 




14^ degrees 


4 
5 


I Pitch cone angle — d 


j Pinion 
\ Gear 


15°-48' 
74°-12' 


14°-48' 
75°-12' 


15°-48' 
75°-12' 


6 


Spiral angle — a 







19°-45' 


31°-21' 


7 


[ Pinion thrust j Direct 
I in pounds per { drive 


f Actual 
\ Calculated 


7.34 


-28.70 


-49.50 


8 


7.06 


-27.6 


-50.3 


9 


] 100 pounds of j Reverse 
1 tooth load \ drive 


j Actual 
{ Calculated 


7.62 


45.00 


73.82 


IP 


7.06 


41.6 


66.9 



252. Skew Bevel Gears. — Another form of special bevel gear, 
known as a skew bevel gear, has no common axes plane, and hence 
the face and cutting angles of the pinion and gear do not converge 
to a common apex. This fact introduces more or less involved 
mathematical calculations in arriving at the various angles re- 
quired to lay out such gears. Because of the more involved 
calculations required and the greater cost of manufacture, skew 
bevel gears are rarely used in machine construction. Strictly 
speaking, there are two distinct types of skew bevel gears, as 
follows: (1) Those in which the oblique teeth are confined to the 
gear, and the mating gear or pinion is really a straight tooth 
bevel; (2) those in which the teeth of both gear and pinion are 
oblique. 

References 

American Machinist Gear Book, by C. H. Logtje. 
A Treatise on Gear Wheels, by G. B. Grant. 
Spur and Bevel Gearing, by Machinery. 
Elements of Machine Design, by J. F. Klein. 
Constructeur, by Retjleaux. 

Handbook for Machine Designers and Draftsmen, by F. A. Halset. 
Bearing Pressures Due to the Action of Bevel Gears under Load, Mchy., 
vol. 20, p. 639. 

Gleason Spiral Type Bevel Gear Generator, Mchy., vol. 20, p. 690. 
Spiral Type Bevel Gears, Mchy., vol. 23, p. 199. 
Laying out Skew Bevel Gears, Mchy., vol. 23, p. 32. 



CHAPTER XIV 

SCREW GEARING 

The term screw gearing is applied to all classes of gears in 
which the teeth are of screw form. Screw gearing is used for 
transmitting power to parallel shafts as well as to non-parallel and 
non-intersecting shafts. The following two classes of screw 
gearing are used considerably in machine construction: (a) 
helical gearing; (6) worm gearing. 

HELICAL GEARING 

253. Types of Helical Gears. — Helical gearing may be used for 
the transmission of power to shafts that are parallel, or to shafts 





Fig. 174. 

that are at right angles to each other and do not intersect, or to 
shafts that are inclined to each other and do not intersect. The 
teeth of helical gears used for connecting shafts that are parallel 
have line contact, while those used for connecting non-parallel, 
non-intersecting shafts have merely point contact and for that 
reason are not used much for the transmission of heavy loads. 
From Fig. 174, it is evident that the normal component of the 
tangential load W on the teeth of a pair of helical gears connect- 
ing two parallel shafts produces an end thrust on each shaft. To 

350 



Art. 254] 



HELICAL GEARING 



351 



overcome this objectionable end thrust, two single helical gears 
having teeth of opposite hand are sometimes bolted or riveted 
together, forming what is called the double-helical or herringbone 
gear. Due to improved methods of cutting helical teeth, herring- 
bone gears are not now constructed to any great extent from two 
single-helical gears, but are cut directly from the solid blank. 
Herringbone gears are also produced by casting them in a prop- 
erly constructed mould. 

There are two general types of double-helical gears, as follows : 

(a) The ordinary herringbone gear in which the two teeth meet 

at a common apex at the center of the face, as shown in Fig. 

175(a). A modification of this type, in which the central part 

has been removed, is shown in Fig. 175(6). 




(a) 



(b) 
Fig. 175. 



(C) 



(6) The type known as the Wuest gear in which the teeth 
instead of coming together at a common apex at the center of the 
face do not meet at all, but are staggered as shown in Fig. 175(c). 

In the types illustrated by Fig. 175(6) and (c), a groove is 
turned into the face as shown, so as to provide clearance for the 
cutters used in cutting the teeth. In gears having teeth cast 
approximately to shape, the center part where the two teeth come 
together is cast somewhat undersize on both sides of the teeth, 
also at the bottom of the space between the teeth as shown in 
Fig. 175(a). 

254. Advantages of Double-helical Gears. — When com- 
pared with a spur gear, a double-helical gear has the following 
advantages : 

(a) The face of the gear is always made long so that more than 
one tooth is in action; in other words, the continuity of tooth 



352 APPLICATIONS OF HELICAL GEARING [Chap. XIV 

action depends upon the face of the gear and not upon the num- 
ber of teeth in the pinion as with spur gearing. 

(b) Due to the continuity of action, the load is transferred 
from one tooth to another gradually and without shock, thus 
eliminating to a great extent noise and vibration. 

(c) In helical gearing, the load is distributed across the face of 
the gear along a diagonal line, thus decreasing the bending stress 
in the teeth. 

(d) In well-designed helical gearing all phases of engagement 
occur simultaneously, hence the load is transmitted by sur- 
faces that are partly in sliding contact and partly in rolling con- 
tact. Such action has a tendency to equalize the wear all over 
the teeth, consequently the tooth profile is not altered. 

(e) Actual tests on double-helical gears show that they have 
much higher efficiencies than those obtained from spur gears. 
Efficiencies of 98 to 99 per cent, are not unusual with properly 
designed transmissions. 

(/) Gear ratios much higher than those used with spur gearing 
may be employed. 

(g) Due to the absence of noise and vibration, double-helical 
gears may be run at much higher pitch line speeds than is pos- 
sible with spur gearing. 

255. Applications of Double -helical Geafs. — Cut double-heli- 
ical gears have been applied successfully to many different classes 
of service. The following examples of applications give some 
idea of the extent of the field in which such gears may be used. 

(a) Drives for rolling mills. — Gears used for driving rolling mills 
operate under very unfavorable conditions, such as heavy over- 
loads, the magnitudes of which are difficult to determine; further- 
more, these overloads are applied suddenly and are constantly 
repeated. The gears are also subject to excessive wear due to the 
dirty surroundings. Double-helical gears are now installed for 
rolling-mill drives, and, due to the continuous tooth engagement, 
such gears readily withstand the suddenly applied loads. When- 
ever it is possible, the gears should be enclosed by a casing and 
run in oil, thereby eliminating all noise. 

(6) Drives for reciprocating machinery. — Gears for motor-driven 
reciprocating pumps and air compressors are required to transmit 
a torsional moment which varies between rather wide limits, sev- 
eral times per revolution. Due to the load fluctuation, an or- 
dinary spur-gear drive is noisy and is subject to considerable 



Art. 256] HELICAL GEAR TOOTH SYSTEMS 353 

vibration, while a double-helical gear drive runs quietly, without 
vibration, and at the same time is more efficient. 

(c) Drives for hoisting machinery. — In connection with motor- 
driven hoists such as are used in mines, double-helical gears are 
especially well adapted, since the high gear ratios possible sim- 
plify the drives. High-ratio double-helical gears are more 
efficient and run more quietly than spur gears having the same 
ratio. Such high-ratio helical gears are also being introduced on 
modern high-speed traction elevators, with excellent results. 

(d) Drives for machine tools. — Double-helical gears used on 
motor-driven machine tools produce a noiseless drive free from 
vibration, and are better adapted to the high speeds that are now 
common in machine-tool drives. 

(e) Drives for steam turbines. — Gears used for reducing the speed 
of a steam turbine to that required by a centrifugal pump, fan, or 
generator must be made accurately, as the pitch line velocity is 
likely to be from 3,000 to 5,000 feet per minute. Due to the high 
efficiency and quiet running obtainable by the use of double-heli- 
cal gears, the latter are used extensively in steam-turbine drives. 
In such installations the pinions are always made from an alloy- 
steel forging, and after being machined they are heat treated. 

256. Tooth Systems. — Several of the more prominent manu- 
facturers of double-helical gears agree fairly well on the following 
points relating to the proportions of the teeth: 

1. The tooth profile should be formed by a 20-degree involute 
curve, thus making the tooth-pressure angle 20 degrees. 

2. The tooth should be made shorter than the old standard 
used with spur gears. 

3. The angle of the helix, more commonly called the angle of 
inclination of the tooth, should be 23 degrees. 

4. The diametral pitch standard should prevail for all cut teeth. 

5. The unequal addendum system should be used on all pinions 
having few teeth. 

(a) Tooth proportions. — The proportions for the teeth and gear 
blank given in Table 79 are those proposed and recommended by 
Mr. P. C. Day of The Falk Co. of Milwaukee, Wis. It should be 
noted that according to these formulas the pitch and outside 
diameters of gears having less than 20 teeth are made slightly 
larger than those of a standard gear. This is done to avoid under- 
cutting of the teeth. If a pinion proportioned in this way 
meshes with a gear having less than 40 teeth, then the distance 



354 



STRENGTH OF HELICAL TEETH [Chap. XIV 



between the shafts must be increased by an amount equal to one- 
half of the increase in the pinion diameter. If the gear, meshing 
with a small pinion has more than 40 teeth the normal center 

Table 79 

1. Tooth profile Involute. 

2. Pressure angle 20 degrees. 

3. Angle of helix 23 degrees. 

8 

4. Length of addendum = - 1 — = 0.2546p'. 

5. Length of dedendum = — — '= 0.3183 p' . 

6. Full height of tooth = — = 0.5729 p'. 

95 T + 1 

7. Pitch diameter, when T < 20 = — 

m V 

T 

8. Pitch diameter, when T > 20 = — 

V 
95 T 1 2 6 

9. Outside diameter, when T < 20 = 



10. Outside diameter, when T > 20 = 



V 

T + l.( 



Table 80. — Pkoportions op Teeth for 

Cut Double-helical Teeth, 

Fawcus Machine Co. 



distance may be used by decreasing the pitch diameter of this 
gear by the same amount that the pinion diameter was increased. 

Gears made according to 
the above suggestions have 
teeth of standard depth but 
unequal addendums. 

In Table 80 are given the 
commercial pitches, tooth 
proportions, and minimum 
lengths of face recommended 
by the Fawcus Machine Co. 
for double-helical gears hav- 
ing a pressure angle of 20 
degrees and a helix angle of 
23 degrees. 

257. Strength of Double - 
helical Teeth. — Various 
formulas have been pro- 
posed for determining the 
working load that a cut 
double-helical gear will transmit ; probably the most reliable are 
those given by Mr. W. C. Bates and Mr. P. C. Day. 



Pitch 


Adden- 
dum 


Deden- 
dum 


Minimum 


Dia. 


Cir. 


face 


8.00 


0.393 


0.100 


0.125 


2.5 


6.00 


0.524 


0.133 


0.167 


3.5 


5.00 


0.628 


0.160 


0.200 


4.0 


4.00 


0.785 


0.200 


0.250 


5.0 


3.50 


0.898 


0.229 


0.286 


5.5 


3.00 


1.047 


0.267 


0.333 


6.5 


2.50 


1.257 


0.320 


0.400 


7.5 


2.00 


1.571 


0.400 


0.500 


9.5 


1.75 


1.795 


0.457 


0.572 


11.0 


1.50 


2.094 


0.533 


0.667 


12.5 


1.25 


2.513 


0.640 


0.800 


15.0 


1.00 


3.142 


0.800 


1.000 


19.0 



Art. 257] STRENGTH OF HELICAL TEETH 355 

(a) Bates' Formula. — In an article entitled "The Design of 
Cut Herringbone Gears," published in the American Machinist, 
Mr. W. C. Bates, mechanical engineer of the Fawcus Machine 
Co., proposed a formula for the permissible working load for a . 
double-helical gear, which is really an adaptation of the well- 
known Lewis spur-gear formula given in Art. 223. The author 
introduces two additional factors, one of which depends upon 
the condition of the load, whether it is constant or variable, 
and the second takes into consideration the lubrication neces- 
sary to prevent wear. In addition to these factors, higher fiber 
stresses than those commonly used with the Lewis formula 
are recommended. The formula as proposed by Mr. Bates is 
as follows: 

W = y 7 Sp'fy CK, (386) 

in which the factors p', f, and y have the same meaning as assigned 
to them in Art. 223. 

The factor C depends upon the ratio of the maximum load to 
the average load during a complete operating cycle. If the 
load is fairly uniform, that is, if the ratio of maximum to average 
load is practically unity, then C is given its maximum value, 
namely unity. If, however, the load on the gear varies, say 
from zero to a maximum twice in a revolution, as, for example, 
when the gear drives a single-cylinder pump or compressor, 
then C must be given some value less than unity. Experience 
should dictate the magnitude of the factor C, and the following 
values, obtained from information furnished by Mr. Bates, 
will serve as a guide in the selection of the proper value for any 
particular class of service. 

1. For reciprocating pumps of the triplex type, C usually is 
taken as 0.7. 

2. For mine hoists running unbalanced, C is taken as 0.57. 

3. For rolling mill drives in which the flywheels are located on 
the pinion shaft, the factor C varies from 0.50 to 0.66, depending 
upon the rapidity with which the energy in the flywheel is 
given up. 

The factor K depends for its value upon the effectiveness of 
the lubricating system used with the gears; in other words, the 
wearing conditions of the gear depend upon K. When the gears 
are encased so that the lower part of the gear runs in oil, thus 
carrying a continuous supply of oil to the mating pinion, the fac- 
tor K may be assumed as unity. It is claimed that with such a 



356 STRENGTH OF HELICAL TEETH [Chap. XIV 

system of lubrication double-helical gearing may be operated 
successfully at speeds of 2,000 to 2,500 feet per minute. Experi- 
ence seems to indicate that with speeds exceeding 2,500 feet per 
minute considerable oil is thrown off the gears due to centrifugal 
action, and in such installations it is suggested that the oil, under 
a low pressure, be sprayed against the teeth on the entering side 
near the line of engagement. For other systems of lubrication, 
the values of K given in Table 81 are recommended. 

Table 81. — Values op K as Recommended by W. C. Bates 



Types of lubricating systems 



Min. Max. Mean 



Continuous supply of oil 

Thorough grease lubrication 

Scanty lubrication, but frequent inspection . 
Indifferent lubrication 



1 
0.83 
0.80 
0.77 



1 
0.91 
0.87 
0.83 



1 
0.87 
0.835 
0.80 



The permissible fiber stress S may be determined by means of 
the formula 

S==So 1,200 + V (387) 

which is similar in form to the expression given for the safe stress 
in the case of spur gearing. The values of S given in Table 72 
may also be used for this class of gearing. 

The values of the factor y as recommended by Bates are those 
worked out by Lewis for the 15-degree involute teeth. For com- 
mercial pitches and corresponding gear faces, see Table 80. 

(6) Formula for Wuest gears. — In a comprehensive paper 
before the American Society of Mechanical Engineers, Mr. P. 
C. Day of The Falk Co. gave a simple formula for determining the 
safe working load on the teeth of Wuest helical gears. The 
formula is empirical, as it is based upon the results obtained from 
several years of experience with such gears. Using as far as 
possible the notation given in the preceding discussion, the safe 
working load is as follows : 

W = 0.4 Sp'f, (388) 

in which the factor S represents the shearing stress on a section 
taken at the pitch line. This shearing stress varies with the 



Art. 258] 



MATERIALS FOR HELICAL GEARING 



357 



pitch-line speed, as shown in Fig. 176. The length of the total 
face of the gear should be at least five times the circular pitch, 
and for average conditions six times the pitch gives satisfactory 
service. When the gear ratio is high, the face may be made ten 
times the circular pitch, provided the pinion and gear are mounted 
on rigid bearings located close together. 

When the load transmitted by the gears fluctuates from a mini- 
mum to a maximum, as in the case of single-acting pumps and 
mine hoists, the gears should be designed for a load which repre- 
sents an average between the maximum and mean loads. The 
gears used in connection with motor-driven machine tools should 



1500 
^ 1000 

500 




















































1500 
1000 £ 
500 




























































































































-__/ 




Cor 


!Sg; 


















































*£? 


*-C/ 






































































\ 




































































3/ 


te/ 


Co^ 


































































































£r r 














































































































































L& 


•-/■ 


'ron 



































































































































































































































500 1000 1500 2000 2500 

Velocity in ft. per min. 

Fig. 176. 

be designed to transmit a load equivalent to the rated output of 
the motor at a speed which is taken as the mean between the 
maximum and minimum revolutions per minute. The design of 
high-ratio and rolling-mill transmissions must receive special 
consideration, and should be left to the engineers of the company 
that manufacture such gears. 

258. Materials for Helical Gearing. — In general, soft mate- 
rials such as rawhide, fiber, and cloth should never be used for the 
pinion. Some manufacturers do not consider it good practice in 
high-ratio transmissions to use cast iron for cut double-helical 
pinions, claiming that a forged-steel pinion will cost but little 
more, and, due to its better wearing qualities, will give increased 
life to the transmission. When the tooth pressures are moderate, 



358 



HELICAL GEAR CONSTRUCTION 



[Chap. XI 



cast iron or semi-steel is preferred to steel casting for gears of 
large diameter; but when the loads are heavy, steel casting is 
generally more economical. The carbon content of the grade of 
steel casting used ordinarily for gears varies from 0.25 to 0.30 
per cent. When the gear and pinion are both made of steel, the 
best results are obtained by making the pinion of a different grade 
of steel than that used for the gear; for example, with a gear made 
of steel casting having a carbon content of 0.25 to 0.30 per cent., 
the pinion should be made of a 0.40 to 0.50 per cent, carbon-steel 
forging. For high-pitch line velocities, alloy-steel pinions sub- 
jected to a heat treatment are recommended. Frequently the 
pinion teeth are cut integral with the shaft. 



' / jm'/ ////M% W //////// M 




259. Double -helical Gear Construction. — (a) Rim. — For large 
gears, The Falk Co. has found that whenever possible the rim 
should be made solid, and when the diameter of the gear exceeds 
7 feet the hub should be split. The split in the hub should be 
placed midway between two arms; thus when six arms are used, 
as is their usual practice, two of these arms are perpendicular to 
the split. The Falk Co. has found that with this arrangement the 
casting will contract very evenly, so that the rough gear blank on 
leaving the sand is practically round. It is claimed that such a 
construction, when used with eight arms, produces a casting that 



Art. 259] 



HELICAL GEAR CONSTRUCTION 



359 



is distorted. Figs. 177 and 178 show two large gears made of 
steel casting and built by The Falk Co. 




- -"^zi7{psssr^ 



r" 



^\i^=xt^r 



Fig. 178. 




15 "9s 



Large double-helical gears transmitting heavy loads are fre- 
quently made with a steel-casting rim, cast in halves and bolted to 
a cast-iron spider. The rims of such gears are shown in Figs. 



360 



HELICAL GEAR CONSTRUCTION 



[Chap. XIV 



179 and 180, and the cast-iron spider for the latter is shown in 
Fig. 181. In order to relieve the coupling bolts between the rim 
and the spider of all shearing action, large heavy keys are fitted 




Fig. 180. 




between the rim and the arms of the spider. The rim, being made 
in halves, has the joints split parallel to the tooth angle. These 
rim joints should always be located between two teeth as shown 



Art. 259] 



HELICAL GEAR CONSTRUCTION 



361 



in Fig. 182. Joints made in this manner do not weaken the teeth, 
nor do they interfere with the smooth operation of the gear. 
Bolts and shrink links as shown in Fig. 182 are used for fastening 
together the two halves of the rim. 

Another design of a rim joint is shown in Fig. 183, and as in 
the design just described, the steel-casting rim is fastened to a 
cast-iron spider by means of bolts and shrink links. This joint, 
however, differs from the one shown in Fig. 182 in that a tongue 
and groove are used, the tendency of which is to weaken the tooth 
along the joint, as is evident from an inspection of Fig. 183. 




& 






20"— H h — 20" — 



Drill 3s a Spo t Face 



^ 



PJ 



Fig. 182. 



In Fig. 184 is shown an excellent design of a heavy steel casting 
double-helical gear, cast in halves. The joint is made through 
the arms, and a series of studs as shown hold the two halves of 
the gear together. The studs in the arms are fitted accurately 
into reamed holes, while those in the hub and under the rim are 
fitted very loosely, because it is impossible to ream these holes. 
The split in the rim is made between two teeth and parallel to 
the teeth. 

The rim sections in common use are illustrated in the various 



362 



HELICAL GEAR CONSTRUCTION [Chap. XIV 




\ \ < 11 

r6"-p-l6- -pfc- 

^3 



f^J^T—J^ 



i " 



Spot F ace „ 



t-y 



M2- 



L—27 



Fig. 183. 




MT - ! 



Fig. 184. 



Art. 259] 



HELICAL GEAR CONSTRUCTION 



363 



figures mentioned in the preceding discussion. According to 
Bates, the finished rim thickness under the teeth of cut double- 
helical gears may be arrived at by the following empirical 
formula : 



2 1' 

Rim thickness =— + ■■= 

V 2 



(389) 



In Fig. 185 is shown a double-herringbone pinion, the teeth of 
which are cut integral with the shaft. This shaft with the double 
pinion is used for driving two large gears of a rolling-mill drive. 

(6) Arms. — Arms of elliptical cross-section should never be 
used for double-helical gearing for the reason that they lack rigid- 
ity at right angles to the direction of rotation. For gears not 
exceeding 40 inches in diameter, and having a length of face 




Fig. 185. 

approximating one-sixth to one-eighth of the diameter, Bates 
recommends the use of cross-shaped arms. With gears 
having wider faces than those just mentioned, the H-section 
similar to those shown in Figs. 178 and 181 should be used. 
Furthermore, according to the same authority, the face of cut 
gears should never be made less than one-tenth of the pitch 
diameter, if the gear is to possess sidewise rigidity and no vibra- 
tion is to be set up in the transmission. For heavy rolling-mill 
drives, the face of the gears is unusually long, and for such gears 
The Falk Co. recommends the use of double arms of U cross-sec- 
tion. In general, the section of the arms should be made con- 
siderably heavier where they join the hub so as to insure sound 
castings. 

260. Mounting of Double -helical Gears. — Due to the high 
speeds at which double-helical gears are used, the frames and 



364 CIRCULAR HERRINGBONE GEARS [Chap. XIV 

bearings supporting such gears must be made heavy and rigid. 
The shafts must all be in true alignment, and the pinion and gear 
must have the supporting bearings located close up to the hubs. 
The gear with its mating pinion should be aligned correctly so as 
to eliminate all end thrust. Means for lubricating the trans- 
mission must be provided, and the whole arrangement should be 
made accessible for inspection. For a high-ratio transmission 
running at a high rotative speed, the pinion is generally integral 
with its shaft, and the latter is driven by the prime mover or mo- 
tor through the medium of a flexible coupling. 

261. Circular Herringbone Gears. — Several years ago, the R. 
D. Nuttall Co. developed and introduced a new form of generated 
tooth gear to which the term circular herringbone was applied. 
Such a gear has continuous teeth extending across its face in the 
form of circular arcs. The teeth are generated by two cutters, 
one for each side of the tooth. The profile of these cutters is an 
involute rack tooth, and the pressure angle for the middle section 
of the gear tooth is 20 degrees. This angle, however, varies 
slightly for all the other sections of the tooth, increasing as the 
sections approach the end of the gear face. The Nuttall Co. has 
adopted as a standard for these gears a short tooth having the 
following proportions: 

1. The tooth profile is made a 20-degree involute. 

2. The length of the addendum is made 0.25 p' . 

3. The clearance is made 0.05 p' . 

4. The whole depth of the tooth is made 0.55 p' . 

5. The radius of curvature of the tooth and that of the face of 
the gear are made equal, and should never be less than twenty- 
four divided by the diametral pitch. 

According to the manufacturers, the circular herringbone gears 
have all the advantages of double-helical gears, and in addition 
two special advantages are claimed. 

1. Due to the fact that the tooth is continuous and not grooved 
at the center, it is stronger and at the same time the rim is re- 
inforced. 

2. The lubrication is applied more readily, since the curved 
tooth acts like a cup. 

WORM GEARING 

The type of screw gearing commonly called worm gearing is 
used for transmitting power and obtaining high speed reductions 



Art. 263] HINDLEY WORM GEARING 365 

between non-intersecting shafts making an angle of 90 degrees 
with each other. There are two classes of worm gearing in 
common use, each of which possesses certain advantages over 
the other. 

262. Straight Worm Gearing. — The class of worm gearing 
most frequently used is that in which the worm is straight or of a 
cylindrical shape. The threads of such a worm have an axial 
pitch that is constant for all points between the top and the root 
of the threads. Strictly speaking, there are two types of straight 
worm gearing. In the first of these types, generally called the 
ordinary worm and gear, the hob used for machining the worm 
gear is of constant diameter and is fed radially to the proper 
depth into the blank, both hob and blank being rotated in correct 
relation to each other. The teeth produced are not theoretically 
correct in shape. In place of a cylindrical hob, one that tapers 
may be used, and by feeding it into the gear blank longitudinally 
at right angles to the axis of the blank instead of radially as in the 
preceding case, the worm gear produced has teeth that approach 
very closely the theoretical form. Gears cut by the latter method 
have given much better service and higher efficiencies than similar 
gears cut by the first method. 

Due to the higher grade of product obtained by the use of a 
taper hob, the second type of worm and gear is employed to a 
considerable extent in the rear axle drives of auto-trucks and mo- 
tor cars. The efficiency and load-carrying capacity are practi- 
cally the same as for the hollow-worm type of gearing described 
in the following article. 

263. Hindley Worm Gearing. — In the second class of worm 
gearing, the worm has a shape similar to that of an hour glass. 
It was introduced by Hindley in connection with his dividing 
engine, and worms having a hollow fS.ce are generally called 
Hindley worms. As may be seen from Fig. 186, the worm is made 
smaller in the center than at its ends, so that it will conform to 
the shape of the gear. Since there is a larger contact surface 
between the mating teeth than in the straight worm class, the 
wear is reduced and it is possible to use a smaller pitch and face 
of gear for a given transmission. In the Hindley worm the axial 
pitch varies at every point, since the angle of the helix changes 
constantly throughout the length of the worm. At the center 
of the worm, the helix angle is much greater than at the ends, 
as is evident from an inspection of Fig. 186. 



166 



HINDLEY WORM GEARING 



[Chap. XIV 



Hindley worm gearing is produced by the nobbing process, 
but since the shape of the worm is made to conform to the cir- 
cumference of the gear, it follows that such worms are not 
interchangeable. In other words, a worm intended for a gear 
containing 36 teeth of a given pitch will not mesh correctly 
with a gear having 54 teeth of the same pitch. In order to ob- 
tain good results with the use of Hindley worm gearing, the 
following requirements must be met: 

1. The center distance between the worm and gear must be 
exact. 




2. The center of the worm must conform exactly with the 
center of the gear so as to avoid any longitudinal displacement of 
the worm. 

3. The worm axis must be in proper alignment, relative to the 
gear. 

Experiments conducted on well-designed and properly mounted 
worm gears, as used in motor-car work, show that the efficiency 
and load-carrying capacity of the hollow worm are slightly greater 
than those obtained by means of the straight worm, although the 
difference is small. 

264. Materials for Worm Gearing. — In general, a worm gear 
transmission gives satisfactory service when the worm is made of 
a low-carbon steel and the gear of a good grade of bronze. The 



Art. 264] 



MATERIALS FOR WORM GEARING 



367 



steel for the worm should have a carbon content that will permit 
of heat-treatment without producing serious distortion of the 
worm. The heat-treatment that is generally used is one of 
carbonizing or case-hardening. For this purpose some manu- 
facturers prefer a nickel steel with a low carbon content, while 
others specify an open-hearth high-carbon steel. In Table 82 
are given six different gear bronzes that the Wm. Cramp and 
Sons Ship and Engine Building Co. has found to be satisfactory 
for the various classes of service indicated. 







Table 82. — Cramp's 


Gear Bronzes 








Wt. 
per 
cu.in. 




Permis- 




Bronze 

No. 


Tensile 

strength 


Elastic 
limit 


Range of 
load 


sible 

r.p.m. of 

worm 


Class of service 


1 


40,000 


20,000 


0.316 


not over 1,500 


1,500 


Light loads and high speeds. 


2 


40,000 


20,000 


0.319 


3,000 to 4,000 


1,000 


Moderate loads and speeds. 


3 


45,000 


22,000 


0.321 


3,000 to 4,000 


1,000 


Moderate loads and speeds 
when excessive wear is ex- 
pected. 


4 


30,000 


15,000 


0.300 


5,000 to 25,000 


200 to 400 


For continuous moderate 
loads with intermittent 
heavy load. 


5 


35,000 


18,000 


0.325 


3,000 
1,000 to 1,500 


200 
600 to 900 


For average running condi- 
tions of light loads and mod- 
erate speeds with heavy 
starting torque. 


Parson's 


65,000 


30,000 


0.305 


10,000 to 50,000 


200 


For heavy loads and slow 


Man. 












speeds under excessive strain 


Bronze. 












and shock. 



From the preceding statements it should not be understood 
that steel and bronze are the only materials that are satisfactory 
for worm gearing. A carbonized steel worm and a gear made of 
a high-grade semi-steel casting will give good service for moderate 
loads and speeds. For light loads and low speeds, a carbonized- 
steel worm with a gear made of close-grained cast iron will prove 
satisfactory. 

265. Tooth Forms. — (a) Straight worm. — The standard form 
of tooth used for the ordinary worm gearing is that proposed 
and adopted as a standard by the Brown and Sharpe Mfg. Co. 
As shown in Fig. 187, the sides of the worm thread make an angle 
of 29 degrees with each other, or in other words, the pressure 
angle is 14J^ degrees. This form of worm thread is produced 
by a straight-sided tool having fiat ends, and for the various 
pitches in use, the proportions may be taken from Table 83. 



368 



TOOTH FORMS FOR WORM GEARING [Chap. XIV 



Table 83.— Standard 29° Worm Threads 



The teeth on the gear which mesh with a worm having teeth 
according to the proportions shown in Table 83 are given an 

involute form, and, 
according to the 
Brown and Sharpe 
Mfg. Co., such gears 
should always have 
more than 31 teeth 
in order to avoid 
undercutting of the 
teeth. 

In modern manu- 
facturing, the so- 
called straight 
worms are no longer 
turned on a lathe, 
but are milled. 
With the use of the 
29-degree thread, 
there is some difficulty in milling such a worm when the helix 
angle approaches 28 degrees. To obviate any difficulty that 





Threads 
per 
inch 


Tooth 

height 

above 

pitch line 


Total 

height 
of tooth 


Width of tooth at 


pitch 


Top 


Bottom 


H 


4 


0.0796 


0.1716 


0.0838 


0.0775 


Vi 


sy 2 


0.0909 


0.1962 


0.0957 


0.0886 


H 


3 


0.1061 


0.2288 


0.1117 


0.1033 


H 


2V 2 


0.1273 


0.2746 


0.1340 


0.1240 


V 2 


2 


0.1592 


0.3433 


0.1675 


0.1550 


2 A 


iy 2 


0.2122 


0.4577 


0.2233 


0.2066 


3 A 


lVs 


0.2387 


0.5150 


0.2512 


0.2325 


l 


i 


0.3183 


0.6866 


0.3350 


0.3100 


IK 


y 5 


0.3979 


0.8583 


0.4187 


0.3875 


iy 2 


% 


0.4775 


1.0299 


0.5025 


0.4650 


m 


Yi 


0.5570 


1.2016 


0.5862 


0.5425 


2 


y 2 


0.6366 


1.3732 


0.6708 


0.6200 




may arise, the angle between the sides of the tooth is made larger 
than 29 degrees. Some designers have adopted an angle of 60 
degrees, while others vary the angle for different helix angles. 



Art. 266] LOAD CAPACITY OF WORM GEARING 369 

(b) Hindley worm. — According to the practice of the Keystone- 
Hindley Gear Co., the angle included between the sides of the 
teeth varies considerably, as is shown by the following: 

1. For single-threaded worms, the angle is made 29 degrees. 

2. For double-threaded worms, the angle is made 35 degrees. 

3. For triple-threaded worms, the angle is made 35 degrees. 

4. For quadruple-threaded worms, the angle is made 37^ 
degrees. 

5. For worms of small diameter having from two to four 
threads, the tooth angle is made as high as 52 degrees. 

Furthermore, this same company has no uniform depth of 
tooth, as it varies from 75 to 100 per cent, of the circular pitch, 
with an average of about 85 per cent. , 

In the Lanchester worm gearing, which is probably one of the 
most efficient types of Hindley gearing in use, the side of the tooth 
is given a slope of 1 in 2. 

266. Load Capacity. — The permissible load upon the worm- 
gear teeth depends more upon the heating effect and wear pro- 
duced than upon the strength of the teeth. If the oil film be- 
tween the teeth in contact breaks down, due to high pressure 
or to thinning of the lubricant caused by high temperatures, 
excessive heating and wear will result. If not remedied, this 
will in a short time destroy the gear or worm, or both. The 
formulas in use for determining the permissible load on worm- 
gear teeth are all of an empirical nature, having the following 
form: 

W = Cfp', (390) 

in which / and p' denote the face and circular pitch, respectively, 
and C is a coefficient depending upon the speed, pressure, and 
temperature. This coefficient must be determined by means of 
experiments. 

In 1902, Prof. C. Bach and E. Roser made an experimental 
investigation of a triple-threaded soft-steel worm and bronze 
worm gear running under various conditions. The pitch diam- 
eter of the worm was a trifle over 3 inches and the lead was 3 
inches, thus giving a helix angle of 17 degrees 34 minutes. 
The worm gear contained 30 teeth of involute profile having a 
pressure angle of 14^ degrees. The results of these tests were 
published in the Zeitschrift des Vereins deutscher Ingenieure 
of Feb. 14, 1903, also in the American Machinist, July 16 and 23, 



370 STRENGTH OF WORM-GEAR TEETH [Chap. XIV 

1903. The expression for the allowable load on the worm drive 
as proposed by Bach and Roser is more or less involved, and 
since it is based upon the investigation of a single worm trans- 
mission, its adoption as a working formula may be questioned. 
The Bach and Roser formula, assuming continuous service, is 
as follows: 

W = (mt + n)/y, (391) 

in which /' denotes the face of the worm gear measured in inches 
on an arc at the base of the teeth; p' denotes the divided pitch 
of the worm or the circular pitch of the worm gear; t denotes the 
rise in degrees F. in the temperature of the oil in the reservoir; 
m and n are experimental coefficients depending upon the velocity 
of the teeth. The relations existing between the velocity V 
in feet per minute and the coefficients m and n are given by the 
following expressions: 

934 



305,520 .__ 
7+^42 " 356 



(392) 



For ordinary working conditions, the temperature rise t in 
(391) may be assumed to vary from 80° to 100°F. If the drive 
is to be installed in a place where the prevailing temperature is 
high, the magnitude of t should be based upon the temperature 
at which the lubricant used in the drive loses its lubricating 
qualities. In view of the fact that formula (391) is based upon 
continuous service, it seems reasonable that for intermittent 
service the permissible load as determined by (391) may be in- 
creased; in other words, instead of designing the drive for the 
maximum load, the average load might be used in arriving at the 
safe dimensions of the worm-gear teeth. 

267. Strength of Worm-gear Teeth. — It may occasionally be 
necessary to investigate the teeth of the worm gear for strength, 
and in such cases the formulas derived for spur gearing may be 
used by making the following modifications: 

(a) For cast gearing, the load W should be considered as coming 
upon a single tooth. 

(6) For cut gearing, assume the load W as equally distributed 
among all the teeth in actual contact as given by (408). 

(c) For the magnitude of / in the spur-gear formula, determine 
the actual length of the gear tooth at the base of the tooth. 



Art. 268] FORCE ANALYSIS OF WORM GEARING 



371 



268. Force Analysis of Worm Gearing. — In order to arrive 
at the probable pressure coming upon the various bearings used 
in the mounting of a worm-gear drive, it is necessary to deter- 
mine the relation existing between the turning force on the worm 
and the tangential resistance on the worm gear. Having 
established this relation, the magnitudes of the various com- 
ponents of the tangential resistance may then be determined, 
and from these components the pressures upon the bearings 
may be found. 

(a) Relation between effort and load. — The relation between the 
equivalent turning force P on the worm and the tangential load 
W upon the worm gear may be obtained as follows : 




Fig. 1 



Referring to Fig. 188, the vector N represents the normal 
reaction between the teeth at the point of contact 0. The symbol 
r denotes the pitch radius of the worm; a the angle of the helix 
of the worm; /3 the pressure angle or the angle the side of the 
thread makes with a line at right angles to the center line of the 
worm. The angle p is the angle of friction for the materials in 
contact. 

Disregarding the frictional resistances, the components of the 
normal force N along the X, Y, and Z axes are, respectively, 

N c = Ncos 6 cos a 
N y = Ncos 6 sin a 
N z = Nsin 6 

Assuming that the worm shown in Fig. 188 rotates in the direc- 
tion as indicated in Fig. 188(a), the force fiN due to friction 



372 FORCE ANALYSIS OF WORM GEARING [Chap. XIV 

upon the worm acts along the tangent to the helix. This force 
of friction tends to increase or decrease the components found 
above; hence resolving pN along the X and Y axes, we obtain 
the remaining components: 

N' x = uNsin a 
N' y = juiVcos a 

Each of the five components is shown in Fig. 188. Now 
adding the components along the same lines of action, we obtain 
the following expressions : 

The magnitude of the tangential force exerted by the worm 
gear upon the worm teeth is 

W = N x - N' x = N (cos cos a - n sin a) (393) 

The magnitude of the turning force P required at the pitch 
radius of the worm is obtained by adding N v and N' v thus 

P = N y + N' y = N (cos sin a + p cos a) (394) 

The force S, causing a downward pressure upon the worm shaft 
or an upward pressure upon the worm-gear shaft, has a magni- 
tude given by N z above, namely, 

S = N z = N sin e (395) 

The relation between P and W may now be obtained by com- 
bining (393) and (394); thus 

p = Tr r cos sin a + M cos on (3Q6) 

Lcos 6 cos a — jj. sin a\ 

Denoting the ratio of n to cos by tan p', (396) reduces to a sim- 
ple form of expression which is similar to that derived for screws, 
namely, 

P = W tan(a + <p') (397) 

Letting p' denote the lead of the worm and writing \j! = tan <p', 
(397) may be put into the following form: 

P = W\^^4^} (398) 

L2 7rr — im'p'J 

From Fig. 188, it is evident that 

tan = tan /3 cos a 

Hence M ' = ( ^ = M Vl + cos 2 a tan 2 (399) 



Art. 269] 



BEARING PRESSURES 



373 



Now fx' may be considered a "new coefficient of friction" peculiar 
to worm gearing and its magnitude may be obtained by means 
of (399). 

Combining (393) and (395) and reducing to a simple form, the 
magnitude of the force S in terms of W is given by the following 
expression : 

(b) Efficiency of worm gearing. — An expression for the efficiency 
of a worm and gear may now be determined. In the ideal trans- 
mission, namely, one having all of the frictional resistances elimi- 
nated, if is apparent that the effort P required at the pitch 
radius of the worm is as follows : 

Po = W tan a (401) 

Hence the efficiency of the worm and gear, not taking into consid- 
eration the frictional resistances of any of the bearings used in 
the mounting, is given by the following formula: 

Po tana . . 

" = F " tan (« + <T> (402) 

269. Bearing Pressures. — (a) Worm shaft. — The worm shaft 
is generally supported on two bearings, each of which must be 




Fig. 189. 

capable of withstanding the pressure coming upon it due to the 
forces P, W, and S. In addition to the transverse forces, the 
worm shaft is also subjected to a thrust, and for that reason 
a thrust bearing must be provided. When ball bearings are used 
for mounting the worm shaft, it is possible to select a type of 
radial bearing that is capable of taking care of a certain amount 
of end thrust in addition to the transverse load. Such a bearing 
makes the installation of a special thrust bearing unnecessary. 
In Fig. 189 is shown a worm shaft mounted on radial bearings 



374 



BEARING PRESSURES 



[Chap. XIV 



that are capable of taking an end thrust equivalent to one and 
one-half times the radial load. Assuming that the turning force 
P and the downward pressure S are applied midway between 
the bearings A and B, each of these bearings is subjected to a 
pressure equal to one-half of these forces. Since S is at right 

.angles to P, the compo- 
nents of these forces at 
the bearings are at right 
angles to each other. The 
tangential force W, in ad- 
dition to causing an end 
thrust upon the bearing A, 
also produces a pressure 

Wr 
equal to —j- upon each 

bearing, the one at A act- 
ing downward and that at 
B upward. Hence the re- 
sultant pressure upon the 
bearing A is as follows : 




IP 2 , \S . WrV 

Vt + L2 + t\ 



(403) 



and the resultant pressure 
upon the bearing B is 



Fig. 190. 



B 



-4 



vWr 
L I 



(404) 



Having determined the magnitudes of the bearing pressures and 
thrusts, the size of bearing may now be selected from tables 
furnished by the manufacturers of such bearings. 

(b) Worm-gear shaft. — The pressure exerted upon the bearings 
supporting the worm gear depend upon the magnitudes of the 
forces P, W, and S, as well as upon the method of mounting the 
gear. The transmission illustrated in Fig. 190 has the gear sup- 
ported on ball bearings mounted on the extended hubs of the 
gear. In some installations, the gear is keyed to a shaft which 
in turn is supported on proper bearings. If the bearings C and 
D in Fig. 190 are located symmetrically with respect to the center 
plane of the worm gear, the pressures upon them due to the forces 



Art. 270] WORM AND GEAR CONSTRUCTION 375 

S and W will be equal to one-half of these forces. The force P 
tends to move the gear along its axis, thus producing a thrust on 
the bearing D, and at the same time this force introduces a trans- 
verse pressure upon both of the bearings. The transverse pres- 

pp 
sures due to P have a magnitude — > the one acting upward 

on the bearing C and the other downward on the bearing D. 
Since P causes an end thrust, it is necessary that the radial ball 
bearings used for supporting the gear be of a type that is capable 
of supporting a thrust in addition to the radial load. Proceeding 
as in the case of the worm shaft, the following expressions are 
obtained : 
The resultant radial load on the bearing C is 



and the resultant radial load on the bearing D is 



(405) 



* = V ] 



|+v] ■ < 406 > 



270. Worm and Gear Construction. — In many worm gear 
transmissions, the worm is made integral with the shaft as shown 
in Figs. 186 and 194 to 197, inclusive. However, occasionally in 
machine tools using worm drives, it is desirable to make the worm 
separate from the shaft and fasten it to the latter by means of 
keys or taper pins as shown in Figs. 187 and 191. 

Worm gears made of cast iron, semi-steel, or steel casting are 
constructed in the same way as ordinary spur or helical gearing. 
If the gear is relatively small the solid or web construction shown 
in Fig. 191 is used. With gears of large diameter considerable 
material may be saved by the use of arms in place of a web. 
The dimensions of the arms may be determined by the formulas 
given in Art. 229. When bronze is used for the gear the cost 
may be kept down by making the rim of bronze, as shown in 
Fig. 186, and bolting it to a spider made of cast iron, semi- 
steel, or steel casting. An example of a worm gear having a 
bronze rim bolted to a cast-iron spider is shown in Fig. 197. 

(a) Length of worm. — In the worm and gear, shown diagram- 
matically in Fig. 192, the symbol D denotes the pitch diameter of 
the gear, and a the addendum of the teeth. The intersections 
of the addendum line of the worm with the addendum circle 



376 



WORM AND GEAR CONSTRUCTION [Chap. XIV 



of the gear are the extreme points of available tooth contact; 
thus the chord AB represents the minimum length of the straight 



V \\ * 


^ 


y 










WlSk 




















^^^^^^^ 














T3 


















3 

J 


1 






w 


iff 




3 


1 — 


J Vi VJ 


f V II V ! 








Fig. 191. 



type of worm in order that complete tooth action may be 
obtained. The expression for the length AB is as follows : 



A = 2\/2aD = (D + 2q) sin d 



(407) 



For worms of the Hindley type, 
the length as recommended by 
Lanchester is such that the differ- 
ence between the maximum and 
minimum diameters is approxi- 
mately 7 to 8 per cent, of the latter. 

Having determined the length 
of the chord AB by means of (407), 
the number of gear teeth in actual 
contact with the worm is then 
given by the formula 

A R 

T 1 = ~ (408) 

(b) Face of the gear. — The face of the worm gear depends upon 
the included face angle of the worm. In Figs. 191 and 193 are 
shown two ways of making the face of worm gears. The design 




Fig 192. 



Art. 271] 



WORM AND GEAR CONSTRUCTION 



377 



shown in Fig. 191 is used considerably for all ordinary worm gears. 
The face angle 25 is chosen arbitrarily, and 60 degrees seems to 
answer very well for all common proportions, although occasion- 
ally 75 degrees may be preferred. 

The large diameter D 2 of the gear blank is given by the follow- 
ing expression, provided the corners of the teeth are left sharp:. 

D 2 = D y + (d - 2 a) (1 - cos 5) (409) 

in which Di denotes the so-called throat diameter and is equal 
to the pitch diameter D plus twice the addendum of the worm 
teeth. 

The design illustrated by Fig. 193 is intended chiefly for worm- 
gears having a large angle of lead. According to the practice 
of one manufacturer of 
such gears, the magnitude 
of the face angle 25 may 
be obtained from the for- 
mula 



cos 5 



3a 



(410) 




Fig. 193. 



in which d denotes the 
pitch diameter of the 
worm, as shown in the 
figure. The outside diam- 
eter D 2 of the gear blank 
•represented in Fig. 193 is 
made equal to the pitch diameter plus three times the adden- 
dum. The throat diameter Di is made equal to the pitch diam- 
eter plus twice the addendum. 

271. Sellers Worm and Rack. — On planers and large milling 
machines, the table is driven by a worm and rack. The teeth 
of the rack are cut straight across and not at an angle; hence the 
axis of the worm must be set over through an angle equal to the 
helix angle. The worm runs in an oil bath and proper thrust 
bearings are provided to take care of the thrust in either direction. 
This form of worm and rack drive was introduced by the Wm. 
Sellers Co. on its planers and later on it was adopted by several 
manufacturers of large milling machines. 

272. Worm-gear Mounting. — Generally speaking, all worm- 
gear transmissions should be mounted in a dustproof casing 



378 



WORM-GEAR MOUNTING 



[Chap. XIV 



which permits either the worm or the gear to run in an oil bath. 
In many installations the worm is located below the gear, while 
in others it must be located above. In the former case the worm 
runs in oil, and experience seems to indicate that such a mounting 
gives the least trouble and lasts longer than the second type. 
There are, however, many installations in which the worm must 
be mounted above the gear, and in such cases the proper lubrica- 
tion depends upon the amount of oil carried to the worm by the 
gear, the lower segment of which runs in the oil bath. Many 
such drives, provided with the proper kind of a lubricant, are in 
successful use. 

From the discussion of the various forces acting upon the 
several elements of a worm-gear drive, it is evident that the 
thrust along both the worm and worm-gear shafts must be taken 




Fig. 194. 



care of by suitable thrust bearings. Figs. 189, 190, and 194 to 
196, inclusive, show several ways of taking care of the thrusts" 
upon the shafts of a worm-gear transmission. In a drive in 
which the efficiency is low or of little consequence, the thrust 
along the worm shaft is taken up by one or more loose washers 
made of bronze or fiber. If more than one washer is necessary, 
then alternate washers of steel and bronze give satisfactory 
service. The shaft bearings of a drive of this kind are generally 
made of bronze, but a good grade of babbitt may also be used. 
On the worm-gear shaft bronze or babbitted bearings may be 
employed, depending upon the magnitudes of the loads coming 
upon the bearings. 

In a drive in which the efficiency must be made as high as 
possible, ball or roller bearings must be used. ' In Figs. 194 and 
195 are shown two examples of a motor-truck rear-axle worm 
mounting in which ball bearings are used. The end thrust upon 



Art. 272] 



WORM-GEAR MOUNTING 



379 



the worm shaft, in the design illustrated by Fig. 194, is taken by 
the double-row ball bearing, and, at the same time, this bearing 
takes its share of the transverse loads upon the shaft. The 
double-row ball bearing is mounted rigidly as shown, while the 
single-row bearing has its outer race floating, thus making pro- 




Fig. 195. 

vision for expansion of the worm shaft. The design just de- 
scribed was originated by The New Departure Mfg. Co. 

The worm-shaft mounting illustrated by Fig. 195 employs the 
type of radial ball bearing that is capable of taking a thrust, 
the magnitude of which is equal to or greater than the radial 




load coming upon them. Another feature worthy of attention 
is the fact that the worm shaft is always in tension, no matter in 
which direction the thrust of the worm gear acts. 

In Fig. 196 is shown another good example of a rear-axle worm- 
gear transmission, in which Timken conical roller bearings are 



380 



TANDEM WORM GEARS 



[Chap. XIV 



used throughout. An inspection of the figure shows that the 
worm shaft is always in compression, and with the rigid mount- 
ing of the roller bearings on this shaft, no provision is made for 
taking care of any expansion that may occur. A mounting simi- 
lar to that shown in Fig. 195, but using conical roller bearings in 
place of the ball bearings, will prove satisfactory. Not infre- 
quently, the worm shaft is mounted upon ordinary radial ball or 
roller bearings and the thrust is taken by a double-thrust ball 
bearing. A combination of radial and thrust bearings is efficient, 
but is more or less complicated and at the same time is more 
expensive than the mountings discussed above. 




273. Tandem Worm Gears. — In heavy-duty elevators, the 
drum or traction sheave is driven by means of double worm gear- 
ing, the arrangement of which is shown in Fig. 197. Such a drive 
consists of right- and left-hand worms cut integral with the shaft 
and mounted below the bronze worm gears with which they mesh. 
The worm gears are, strictly speaking, helical gears and since they 
are cut right and left hand of the same pitch, they readily engage 
with each other. One of these worm gears is connected to the 
hoisting drum or sheave. It is evident that a combination of 
this description practically eliminates all end thrust on the worm 
shaft, thus simplifying the arrangement of the bearings on this 



Art. 274] EXPERIMENTAL RESULTS ON WORM GEARING 381 

shaft. The part of the shaft between the worms is subjected 
either to a tension or a compression, depending upon the loading 
on the hoisting drum. 

274. Experimental Results on Worm Gearing. — A consider- 
able number of tests of worm gearing have been made by various 
investigators in order to determine the probable efficiency of such 
gearing, also to determine the relation existing between the coeffi- 
cient of friction and the sliding velocity of the teeth in contact. 
Evaluating equation (402) for a given coefficient of friction and 
various angles of lead, it will be found that the efficiency varies 
but little for angles between 30 and 60 degrees. The results ob- 
tained from the well-known experiments on worm gearing made 
by Wilfred Lewis agree very closely with those determined by 
means of (402). 

The value of the coefficient of friction for any particular condi- 
tion of speed and tooth pressure is somewhat difficult to deter- 
mine. The experimental results obtained by Lewis, Stribeck, 
Bach, Roser, and other investi- 
gators seem to lead to the Tabl ® 8 _ 4 -~ Re ^ s . °*_ Test _ s on 
following conclusions: (1) The 
coefficient of friction appears 
to have its greatest value at 
low speeds, also at high speeds. 
(2) The coefficient of friction 
has its lowest values at me- 
dium speeds (200 to 600 feet 
per minute). (3) The coeffi- 
cient of friction varies but 
little for different tooth pres- 
sures. In Table 84 are given 

some of the results obtained by Stribeck from a series of tests 
on a cast-iron worm and gear having the following dimensions : 
The gear was 9^ inches in diameter and had 30 teeth. The 
outside diameter of the single-thread worm was approximately 
3% inches, and the tangent of the helix angle was given as 0.1. 

In the design of high-efficiency worm gearing as used in motor 
cars, one authority recommends that n may be taken as low as 
0.002; however, this value appears rather low for general use and 
it is believed that 0.01 will give safer results. For designing 
single-thread worms of the irreversible or self -locking type, the 
coefficient of friction may be assumed as 0.05. 



Cast-iron Worm Gearing 
by Stribeck 



Velocity, 
ft. per min. 


Pressure, 
pounds 


Coef. of fric- 
tion at 60°C. 


98 


| 


0.061 


196 


1,100 


0.051 


294 


j 


0.047 


392 


880 


0.040 


586 


550 


0.030 


784 


350 


0.025 



382 REFERENCES [Chap. XIV 

The actual efficiencies of well-constructed and properly- 
mounted worms and gears, as used on motor cars, are in general 
high, running above 95 per cent, in many cases. 

References 

American Machinist Gear Book, by C. H. Logue. 

Spiral and Worm Gearing, by Machinery. 

Elements of Machine Design, by W. C. TJnwin. 

Worm Gearing, by H. K. Thomas. 

Herringbone Gears, with special reference to the Wuest System, Trans. 
A. S. M. E., vol. 33, p. 681. 

The Design of Cut Herringbone Gears, Amer. Mach., vol. 43, pp. 901 and 
941. 

Power Transmitted by Herringbone Gears, Mchy., vol. 19, p. 782. 

Theory of Enlarged Herringbone Pinions, Mchy., vol. 23, p. 401. 

The Transmission of Power by Gearing, Ind. Eng'g and Eng'g Digest, vol. 
14, p. 114. 

A New Gear — The Circular Herringbone, Amer. Mach., vol. 39, p. 635. 

Making Worm Gears in Great Britain, Amer. Mach., vol. 36, p. 739. 

Manufacturing Hindley Worms, Amer. Mach., vol. 41, p. 149. 

Manufacture of Worm Gearing by a New Process, Trans. Soc. of Auto. 
Engr., January, 1915. 

Gear for Panama Emergency Gates, Amer. Mach., vol. 37, p. 239. 

Allowable Load and Efficiency of Worm Gearing, Mchy., vol. 17, p. 42. 

Experiments on Worm Gearing, Trans. A. S. M. E., vol. 7, p. 284. 

Worm Gear, London Eng'g, Aug. 20 and 27, and Sept. 3, 1915. 

Worm Gear and Worm Gear Mounting, Inst, of Auto. Engr., December, 
1916. 



CHAPTER XV 
COUPLINGS 

A coupling is a form of fastening used for connecting adjoin- 
ing lengths of shafting so that rotation may be transmitted from 
one section to the other. Couplings may be divided into the 
following general groups : (a) permanent couplings; (6) releasing 
couplings. 

PERMANENT COUPLINGS 

A permanent coupling is generally so constructed that it is 
necessary to partially or wholly dismantle it in order to separate 
the connected shafts. Hence, it is evident that permanent coup- 
lings are only used for joining shafts that do not require frequent 
disconnection. Permanent couplings may be grouped into the 
following classes: 

(a) Couplings connecting shafts having axes that are parallel 
and coincident. 

(6) Couplings connecting shafts having axes that are parallel 
but not coincident. 

(c) Couplings connecting shafts having axes that intersect. 

(d) Couplings connecting shafts having inaccurate align- 
ments. 

COUPLINGS FOR CONTINUOUS SHAFTS 

Some of the requisites of a good coupling for connecting con- 
tinuous shafts are as follows : 

1. It must keep the shafts in perfect alignment. 

2. It must be easy to assemble or dissemble. 

3. It must be capable of transmitting the full power of the 
shafts. 

4. The bolt heads and nuts, keys and other projecting parts 
should be protected by suitable flanges, rims, or cover plates. 

275. Flange Coupling. — One of the most common as well as 
most effective type of permanent coupling for continuous shafts 
is the plain flange coupling shown in Fig. 198. In order to insure 
positive shaft alignment, one shaft should project through its 

383 



384 



FLANGE COUPLING 



[Chap. XV 



flange into the bore of the companion flange. Another effective 
way of accomplishing the same purpose is to allow a part of the 
one flange to project into a recess in the other, as shown in Fig. 
198. The coupling bolts must be fitted accurately, generally a 
driving fit, so that each one will transmit its share of the torsional 
moment on the shaft. The size of the bolts should be such that 
their combined shearing resistance will at least equal the tor- 
sional strength of the shaft. In certain installations requiring 
accurate alignment of the shafts, the flanges of the coupling are 
forced on the shaft and are then faced off in place. 




Fig. 198. 



Analysis of a flange coupling. — A flange coupling may fail 
to transmit the full torsional moment of the shaft from the 
following causes: (1) The key may fail by shearing or by crush- 
ing. (2) The coupling bolts may fail by shearing or by crush- 
ing. (3) The flange may shear off at the hub. 

1. Failure of the key. — To prevent the key from shearing, its 
moment of resistance about the axis of rotation must at least 
equal the torsional strength of the shaft. Using the notation 
given in Art. 93, the relation between the shearing strength of 
the key and torsional moment T according to (104) may be 
expressed as follows: 

H > g (411) 



Art. 275] FLANGE COUPLING 385 

To prevent crushing of the key, the moment of the crushing- 
resistance of the key about the axis of rotation must exceed 
slightly the torsional moment T; whence, from (102) 

u * M (412) 

2. Failure of the bolts. — In a flange coupling located at a con- 
siderable distance from the bearings supporting the shaft, the 
bolts are generally subjected to bending stresses in addition to 
crushing and shearing stresses. It is evident, therefore, that 
couplings should be located near the bearings. In the following 
analysis it will be assumed that the coupling bolts are not sub- 
jected to a cross-bending, but only to shearing and crushing 
stresses. Equating the shearing resistance of the bolts to the 
load coming upon them, we obtain the relation 



«>>/i 



IvneS; (413) 

in which a denotes the diameter of the bolts, n the number of 

bolts used in the coupling, and e the diameter of the bolt circle. 

Instead of failing by a shearing action, the bolts as well as the 

flange may fail by crushing; whence we obtain the relation 

9 T 

af :> -=-^, (414) 

■ ~ neS b v ' 

in which / denotes the thickness of that part of the flange through 
which the bolts pass. 

3. Shearing off of the flange. — The coupling may fail due to the 
shearing of the flange where the latter joins the hub. To prevent 
this failure the moment of the shearing resistance of the flange 
must at least equal the torsional moment transmitted by the 
shaft. Hence, it follows that 

c*f > %* (415) 

in which c denotes the diameter of the hub, and S' s the allowable 
shearing stress in the material of the coupling. 

4. Proportions of flange couplings. — In order that a flange coup- 
ling may transmit the full torsional strength of the shaft to which 
it is connected, the various relations derived above must be satis- 
fied. The analysis of the stresses just referred to is only made in 
special or unusual cases. For the common flange coupling used on 



386 



COMPRESSION COUPLING 



[Chap. XV 



line- and counter-shafts, it is unnecessary to make an investiga- 
tion of the stresses in the various parts, as the proportions of such 
couplings have been fairly well established by several manu- 
facturers. However, no uniform proportions of flange couplings 
have as yet been proposed for adoption as a standard. In Table 
85 are given the proportions of a series of flange couplings recom- 
mended by the Westinghouse Electric and Mfg. Co., and these 
represent good average practice. The dimensions listed in Table 
85 refer to the flange coupling shown in Fig. 198. 

276. Marine Type of Flange Coupling. — The type of flange 
coupling shown in Fig. 199 is used chiefly in marine work where 

great strength and reliability are 
of the utmost importance. The 
fitting of this form of coupling is 
done with considerable care; for 
example, the bolt holes are 
always reamed after the flanges 
are placed together, thus insur- 
ing perfectly fitted bolts, each of 
which will transmit its full share 
of the torsional moment upon 
the shaft. The method of 
analyzing the stresses and arriv- 
ing at the dimensions of the vari- 
ous parts of a marine flange coup- 
ling is similar to that given for the common flange coupling. 

277. Compression Coupling. — (a) Clamp coupling.- — A form 
of coupling used extensively at present on shafts of moderate 
diameter, say up to approximately 5 inches, is shown in Fig. 200. 
It is commonly called a compression or clamp coupling. The 
two halves of the clamp coupling are planed off, and after the 
bolt holes are drilled, the halves are bolted together with strips 
of paper between them and bored out to the desired size. After 
the boring operation, the strips of paper are removed. When 
the coupling is fastened to the shaft, the small opening between 
the two halves, due to the removal of the paper, permits the draw- 
ing up of the bolts, and a clamping action on the shaft is thus pro- 
duced. The square key used in connection with a clamp coupling- 
is generally made straight and is fitted only at the sides. This 
coupling may be put on and removed very easily, and it has no 




Fig. 199. 



Art. 275] WESTINGHOUSE FLANGE COUPLINGS 



387 





d 


oo co eo 


■* 


T* ■* 


CO CO CO GO GO 00 


C 
5 


xx xxx xx x 

i-» iH N <M <N CO ■<* rt< »C> CO N. GO 


a" 
5 


XXXXXX XXXXX 


1 

s 


•* 


to 
X 


tN 

X 


X 


CN 

(M tO \M tD 

X X ih XXX 


Ml 


X 


to 

X 


X 


tO 
5D \rH 
\H \00 r-i\ \t)I \00 

os\ i(J\ ih dS\ t-\ 


CN 


tO 50 CN (N CN 50 50 O (O 
rKr\»\ X XX X X! X X X X 


s 


XXX X XX XXXXXX 


O 


50 

50 CO (N 50 50 \jH 
\H \-H \C0 V* \-H \00 \H\M \00 rn\ \0O 

ec\ cc\ i-\ .-N. w\ co\ i>\ r-\ in\ r-i t-\ 


OS 


IN CN N 

CN 50 \C0 \C0 \C0 50 
V* V* VO \-l M\ lrf\ i-\ \<c y \» \H V( 

r-N |H\ Oi\ IO\ iH iH r-4 l»\ C0\ t-\ iH\ i-K 


00 


5D 50 

•XX?? XX XXXXXX 

rH rH T-l H (NN CO 


t^ 


50 50 5D 

50 \H \jH 50 \iH 50 

\O0 \00 \.H 1H\ \00 \CN lri\ \O0 \00 \H !-f\ \rH 

iH\ P5\ C>\ iH i-K r-K rH co\ in\ C0\ iH a>\ 

rH rH rH r-l N (N (NCOCO^TtHlO 


to 


XX XX xxxxx 

N N M CO t* CO NOOOH^N 


Ifl 


tN <M 50 
CN \M \M \r* 50 50 

x^xxrs x xxxxx 

(M <N <N CO ■* W3 NOOOJOCOO 


■* 


XXXXX XXX X 

<M <N <M CO tH CO N 00 OJ ^ M (fl 


- 


\oo \tN V* \so \mv* \cn V* 
i-\ r-\ crfs. cd\ io\i-N r4\ m\ 
N NM CO ION 00 O <N CO CO © 
rH 1— 1 rH i— 1 <N 


« 


xxx xxx xxxxx 

CO^iC CO 00© C0Tt<COC0r-i«O 

rH rH rH rH rH CN CN 


- 


XXX XXX X X 

lO ©N 00 rHCO CO C5 rH ** GO CO 
rH rH HHNNNM 


1 


3 


^XX 


CO 


00 "* 


lOSlNOOON 



388 



CLAMP COUPLINGS 



[Chap. XV 



projecting parts that are liable to injure workmen. In Table 
86 are given the general dimensions of a series of sizes of the clamp 
coupling illustrated in Fig. 200. 



— 7 



a 



QQQQ 






BQS6 







Fig. 200. 
Table 86. — Dimensions of Clamp Shaft Couplings 



Shaft 


Dimensions 


Diam. 
of bolts 


Key 


diameter 


1 


2 


3 


4 


5 


6 


7 


8 


9 


IKe 


6 

7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
18 
20 


4M 
4% 
5K 
6H 

7% 

8 

8% 

9 

9M 

10K 

UK 

12% 


2K 
3 

3K 

3K 

3% 
4 

4M 
4K 
4M 
5 

5K 
6 

6% 


2% 

3 

3M 

3K 

3% 

4 

4K 

5 


1 

IKe 

1M 
IK 
i% 
i% 

2 

2M 

2M 

2K 

2M 

3 

3M 


1% 
2K 
2% 
2K 

2% 

2% 
3K 


IK 
1% 

2 

2% 
2K 
1% 

IK 
1% 
1% 
1% 

2 
2K 

2K 


15 Ae 


K 
K 


Vs 


he 


1% 
1% 


IKe 


3 A 


Vie 


2Kb 




2Ke 


IHe 


Vs 


He 


2% 


lKe 


K 




2% 

3%6 


'Vie 


3Ke 
3% 


1Kb 


Vs 


3^6 

4Ke 


1Kb 


1 


ls Ae 


4^6 


1%6 


IK 





(b) Nicholson compression coupling. — Another form of the so- 
called compression coupling is shown in Fig. 201. This coupling 
requires no cutting of keyways in the shafts that are to be con- 
nected together. It consists of two flanged hubs having tapered 
bores which do not run clear through the hub, but terminate a 
short distance from the outer end as shown in the figure. Double- 
tapering steel jaws are fitted into the tapered bore and held in 



Akt. 278] 



ROLLER COUPLING 



389 



proper position by the key-seats or slots cut into the end of the 
hub. These jaws are machined on the inner faces to a radius 
a trifle less than the radius of the shaft, thus forming a positive 
grip on the shaft when the two flanges are drawn together 
by the bolts. The adjustment of the coupling is always con- 
centric and parallel. No keys are required, thus saving the cost 




Fig. 201. 

of cutting the key-seat in the shaft and of fitting the key. The 
coupling illustrated in Fig. 201 is manufactured by W. H. 
Nicholson and Co. of Wilkes-Barre, Pa. 

278. Roller Coupling. — In Fig. 202 is shown a form of shaft 
coupling in which steel rollers are used for gripping the shaft. As 




shown in the figure, the coupling consists of a cylindrical sleeve 
with two eccentric chambers on the inside. Each of these 
chambers contains two steel rollers, held parallel to each other by 
a light wire frame. With the rollers located in the largest part of 
the eccentric chambers, the coupling may easily be slipped over 
the end of the shaft. A slight turn of the coupling in either direc- 



390 



OLDHAM'S COUPLING 



[Chap. XV 



tion forces the rollers up the inclined sides of the eccentric 
chamber thereby locking the coupling to the shaft. Since no 
screws, bolts, pins, or keys are used with this coupling, no tools 
are needed in applying it to a shaft. Due to the smooth exterior, 
the roller coupling shown in Fig. 202 insures freedom from acci- 
dent to workmen. 



COUPLINGS FOR PARALLEL SHAFTS 

279. Oldham's Coupling. — When two shafts that are parallel, 
but whose axes are not coincident, are to be used for transmitting 
power, a form of connection known as Oldham's coupling is used. 



Zi 



d 

r_ 



Fig. 203. 

The constructive features of such a coupling are shown in Fig. 
203. It consists of two flanged hubs c and d fastened rigidly to 
the shafts a and b. Between these flanges is a disc e, which 
engages each flanged hub by means of a tongue and groove joint, 
thus forming a sliding pair between them. With this form of 
coupling, the angular velocity of the shafts a and b remains the 
same. 

Parallel shafts may also be connected by two universal joints 
in place of an Oldham's coupling. 



COUPLINGS FOR INTERSECTING SHAFTS 

280. Universal Joint. — For shafts whose axes intersect, a form 
of connection known as Hooke's coupling is frequently used. A 
more familiar name for this coupling is universal joint In Figs. 
204 to 207, inclusive, are shown four types of universal joints. 
The type of joint illustrated by Fig. 204 consists of two U-shaped 
yokes which are fastened to the ends of the shafts that are to be 
connected together. Between these yokes is located a cross- 



Art. 280] 



UNIVERSAL JOINT 



391 



shaped piece, carrying four trunnions which are fitted into the 
bearings on the U-shaped yokes. The joint shown in Fig. 204 
is manufactured by the Baush Machine Tool Co., and is well 









-4- 






C- 




3L J 




^ 


f(\ 




■ 10 




v_ 


yui/fi 




V ! 


^^ 



" ~^^ 


k ^ 


I 


T > '' t 


¥^ 




fO 


UAi 


& 




1 


zrt£ 


r y 


1 


L B - 











Fig. 204. 
Table 87. — Proportions of Bocorselski's Universal Joint 









Dimensions 




Diameters 




Size 


















l 


2 


3 


4 


5 1 6 


7 


8 


9 


10 


Ya 


IK 


% 


K 


Kb 


%2 


%4 


H 


0.076 


0.0465 




A 


m 


% 


% 


• X %4 


Vz2 


%2 


Kb 


0.1065 


0.0595 




K 


2 


1 


H 


% 


%2 


%a 


K 


0.167 


0.096 




A 


2K 


1H 


A 


X %2 


% 


% 


Kb 


%2. 


Ha 




y± 


2^16 


1% 


% 


Y2 


X %2 


Kb 


% 


M 


x Ye± 




A 


3M 


1% 


A 


19 A 2 


X %2 


H 


Kb 


%2 


He 




i 


3% 


1% 


1 


% 


He 


Kb 


K 


Ke 


A 2 


K 


m 


3M 


1% 


IK 


A 


X K6 


% 


A 


% 


K 


A 


1H 


4M 


2M 


IK 


1 


% 


% 


M 


H 


X Y%2 


A 


IK 


4M 


2M 


m 


191 6 


% 


% 


% 


Kb 


% 


A 


2 


5Ke 


2 2 %2 


2 


1Kb 


1%2 


1Kb 


1Kb 


% 


Vie 


A 


2H 


7 


3M 


2^ 


1M 


1% 


1Kb 


1%6 


M 


Y2 


A 


3 


9 


4^ 


3 


2^ 


i 2 H 2 


1% 


1Kb 


% 


5 A 


K 


4 10% 


5%6 


4 


2% 


2^6 


2Kb 


1% 


1Kb 


3 A 


K 



adapted for machine-tool service as found on multiple drills. 
In Table 87 are given general dimensions of the Bocorselski's 
patent universal joint shown in Fig. 204. 



392 



UNIVERSAL JOINT 



[Chap. XV 



The coupling shown in Fig. 205 is intended for heavy service, 
as the two yokes and the center cross are made of hard bronze, 
while the screws are made of nickel steel. The maximum 
angular displacement of this joint is limited to 25 degrees. 



a 


"^L. 
















N \ ^^-sJ. 


M 








\ ( A 








: \ % 


m 








/ i ! 
















u 


j±r 





Fig. 205. 

The universal joint in one form or other is used extensively in 
motor-car construction. In Fig. 206 is shown a joint designed by 
the Merchant and Evans Co. of Philadelphia, Pa. The coupling 




Fig. 206. 
Table 88. — Dimensions of Merchant and Evans Universal Joints 



Horse 


Size of 
shaft 


Dimensions 


rating 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


35 
35-80 


1H 

m 


5Vs 
7M 


6 


3 

sy 2 


2Vs 

3y 8 


2% 


1H 
214 


2y 8 
2y 2 


2 

2% 


w 

iy 


3 A 
l 


y 



Art. 280] 



UNIVERSAL JOINT 



393 



consists of a flanged hub to which is attached a ring having radial 
slots. The flanged hub is made of machine steel and the slotted 
ring of a high-carbon steel. Into the radial slots of the ring are 
fitted the projecting arms or teeth of the spider which is also 
made from a high-carbon steel. On the enlargement of the hub 
of the spider is formed a spherical surface which fits accurately 
into a housing, the latter being fastened by bolts to the slotted 
ring and the flanged hub. Spherical centering caps are fitted to 
the inside faces of the flanged hub and spider. All of the spherical 
surfaces have the same center, which, for the design shown, is 
located on the common center line of the two shafts. The maxi- 
mum movement out of true alignment that is permissible with 
the style of coupling shown in Fig. 206 is plus or minus 4 degrees. 
Table 88 gives general dimensions of two sizes of this coupling, 
the smaller of which is capable of transmitting 35 horse power 
and the larger, 80 horse power. 





„EL_ 




IL™ 














K" 1 




In Fig. 207 is shown another design of universal coupling fre- 
quently found on motor cars. The constructive details are shown 
more or less clearly in the figure and hence no further description 
is necessary. 



COUPLINGS FOR SHAFTS HAVING INACCURATE 
ALIGNMENTS 



Frequently it is necessary to connect shafts in which slight 
deviations in alignment must be taken care of, as for example in 
connecting a prime mover to a generator, or an electric motor 
to a centrifugal pump, blower, or generator. For a satisfactory 
connection, flexible couplings are used. Several forms of flexible 
couplings are now used by various manufacturers, and the 



394 



LEATHER-LINK COUPLING 



[Chap. XV 



following are selected as typical illustrations of the different 
types. 

281. Leather-link Coupling. — In Fig. 208 is shown a leather- 
link flexible coupling manufactured by The Bruce Macbeth 
Engine Co. of Cleveland, Ohio. It consists of two flanged hubs 
connected together by leather links as shown in the figure. The 
links are held securely by bolts, which in turn are fastened to the 
flanges so that one end of the links is anchored to the one flange 
while the other end is anchored to the other flange. The torque 
of one shaft is transmitted to the other through the combination 
of flanges, links, and bolts. In order to obtain the desired flexi- 
bility, alternate holes in the flanges are made larger so as to per- 
mit sufficient play for the enlarged washers used on the bolts. 




Fig. 208. 
Table 89. — Data Pertaining to Leather Link Couplings 













Dimensions 






Max. h.p. at 
100 r.p.m. 


Maximum 
r.p.m. 


Weight, 
lb. 








d 
















l 


2 


3 


% 


1.5 


2,400 


15 


5 


5% 


2 


iKe 


2 


2,000 


25 


6 


7 


2K 


1% 


6 


1,800 


65 


8 


10M 


4 


1% 


10 


1,600 


110 


10 


16 


6 


2Ke 


15 


1,500 


210 


13 


20 


8 


2% 


30 


1,250 


. 335 


15 


24 


10 


3Ke 


50 


1,000 


560 


18 


29 


12 


4Ke 


100 


850 


1,270 


26 


34 


14 


Wz 


200 


750 


1,790 


30 


40 


16 



The leather used for the links is made from selected hides and is 
treated by a special tanning process so as to increase the strength 



Art. 2£2] 



LEATHER-LACED COUPLING 



395 



and flexibility. According to one prominent manufacturer of 
leather-link couplings the working stress for the links may be 
taken as 400 pounds per square inch. Due to the low first cost 
of leather-link couplings, the General Electric Co. recommends 
their use on all shafts up to and including 2 inches in diameter. 
For shafts from 2 to 3^ inches in diameter, either the link 
type or the leather-laced type may be used. In Table 89 are 
given general dimensions and other data pertaining to the 
coupling shown in Fig. 208. 

282. Leather-laced Coupling. — The leather-laced flexible coup- 
ling shown in Fig. 209 consists of two cast-iron flanges upon 
which are bolted steel rings. An endless leather belt is laced 




Fig. 209. 



through a series of slots that are formed in the rim of these 
steel rings. The construction used offers a ready means of 
disconnecting the machines without unlacing the belt. As 
may be seen in Fig. 209, disconnection is accomplished by simply 
removing the cap screws that fasten the outer steel ring to the 
central flange. According to the General Electric Co., the manu- 
facturers, this coupling is recommended when the shafts to be 
connected are more than d}4 inches in diameter. The belting 
used is made from a specially prepared leather capable of 
carrying a working stress of 400 pounds per square inch of 



396 



FRANC KE COUPLING 



[Chap. XV 



section. In Table 90 are given general dimensions and other 
data pertaining to the laced-belt coupling shown in Fig. 209. 



Table 90. — Data Pertaining to Leather Laced Couplings 




Max. h.p. 

at 
100 r.p.m. 


Max. 
r.p.m. 


Weight, 
lb. 


Dimensions 


Key 


d 


1 


2 


3 


4 


5 


6 


2>A 


16 


1,200 


160 


15M 


10 


5 


4% 


4»Me 


4?i 


Hxy 2 


3 
3H 


27.7 


900 


263 
256 


18H 


12 


6 


51^6 


5^6 


5Me 


MXM 


4 


66 


750 


494 
482 


24M 


14 


8 


6^6 


^He 


6K 


1X1 


5 


128 


600 


883 

868 


30H 


16 


10 


71^6 


7 1 >f 6 


7H 


1KX1H 


6 


222 


450 


1,329 
1,307 


37 


18 


12 


8^6 


S^Ke 


7% 


1HX1K 


7 
7H 


352 


350 


2,076 
2,046 


43 


20 

24 


14 
16 


9***6 


9% 


8% 


8 

sy 2 


526 


300 


2,767 
2,727 


49 


11^6 


l^He 


9H 


1KX1M 


9 


748 


250 


3.917 
3,865 


55 


28 


18 


13^6 


13% 


9% 


1MX2 


10 


1,027 


200 


5.120 


61 


32 


20 


15^16 


15^6 


10Me 





In general, flexible couplings using leather as the connecting 
medium are not recommended for places where dampness or 
oil would affect the leather. Neither should they be used when 
flying dust or grit are liable to injure the leather links or lacing. 
It is generally assumed that the leather connectors afford suffi- 
cient insulation between the two halves of the coupling when the 
latter is used in connection with electric motors or generators. 

283. Francke Coupling. — The type of flexible couplings dis- 
cussed in the two preceding articles transmit power from the 
driving to the driven member by means of a fibrous material. 
Couplings having soft rubber buffers between interlocking arms 
of two cast-iron spiders have also been used successfully. Re- 
cently a form of coupling known as the Francke flexible coupling 
in which a pair of flanges are connected by flexible steel pins 
was placed on the market. The constructive details of this 
coupling are shown clearly in Fig. 210. The so-called pins are 
built up of a series of tempered-steel plates having a slotted hole 



Art. 283] 



FRANCKE COUPLING 



397 



at each end through which a hardened-steel pin passes. By 
means of these pins, the ends of the tempered plates are held in 
steel yokes which are fastened to the rims of the flanges by means 
of cap screws, as shown in Fig. 210. In the smaller sizes of the 
Francke coupling, the ends of the steel yokes and the inner 
surfaces of the coupling flanges have grooves into which steel 
rings are sprung, thus holding the tempered plates in a radial 
position. 

Any flange coupling connecting two shafts that are out of 
alignment will run open on the one side and closed on the other. 




Fig. 210. 



The endwise motion due to this opening and closing action of the 
flanges is provided for, in the Francke coupling, by the slotted 
holes near the ends of the tempered-steel plates. 

In Table 91 are given general dimensions, net weights, permis- 
sible speeds, and approximate horse powers pertaining to the 
commercial sizes of the coupling shown in Fig. 210. The follow- 
ing directions for selecting the proper size of coupling for any 
desired service are recommended by the manufacturers of the 
Francke coupling. 



398 



FRANCKE COUPLING 



[Chap. XV 



(a) From Table 91, select the smallest coupling having a maxi- 
mum bore large enough to receive the largest shaft to be 
connected. 

(6) For the installation under consideration, determine the 
horse power transmitted per 100 revolutions per minute. 

Table 91. — Data Pertaining to the Fbancke Coupling — Heavy 

Pattern 





Max. 
bore 


Max.h.p. 
at 100 
r.p.m. 


Max. r.p.m. 


Weight 


Dimensions 


Key 


No. 


Cast iron 


Steel 


1 


2 


3 


4 


5 


3H 


% 


1.33 


4,000 


10,000 


8.5 


3X 


iVa 


1^6 


2% 6 


1^2 


M 6 XK6 


4 


l% 


2 


11 


4 


5H 


2<Ke 


2^6 




±x 


m 


2.75 


14 


±X 


5K 


2X 


2% 


1^6 


%x% 


5 


1H 

2 
2X 


3.75 

6.5 

9 


3,500 
3,100 
2,500 


8,500 
7,600 
6,400 


20 
35 
45 


5 
6 

7 




3 

m 


2K 6 

2%6 
2^6 


XeXXe 


7 


%X% 


8y 2 


3 


28 


2,150 


5,400 


70 


&X 


7% 


5X 


3%6 


1% 


XxX 


10 
12 
15 

18 
22 


3H 
4K 
6 

7% 
10 


65 
91 
145 
210 
300 


1,800 
1,500 
1,200 
1,000 
800 


4,600 
3,800 
3,000 
2,500 
2,000 


115 
210 
385 
555 
1,000 


10 
12 
15 
18 
22 


9% 

UH 

13J4 
16% 


&X 

ii 

13% 
17% 


3^6 
4%6 
5% 6 

8% 6 


2% 


%XVs 
%X% 

1XJ4 
IK XI 


24 
27 


9 
11 


750 
1,000 


750 
700 


1,900 
1,700 


1,250 
1,650 


24 
27 


18M 
22H 


16H 

19% 


9 
11 


m 




33 


14 


2,500 


575 


1,400 


3,330 


33 


26H 


24 


13 


5% 





Table 92. — Factors for Various Classes of Service 

Class of service 

Steam turbines connected to centrifugal pumps and blowers 

Turbines and motors connected to generators 

Motors connected to centrifugal pumps and blowers 

Motors connected to wood-working machinery 

Motors connected to grinders, conveyors, screens, and beaters with 

no pulsations 

Motors connected to crushers, tubemills, and veneer hogs 

Gas and steam engines connected to machines carrying a uniform 

load 

Engines connected to fans 

Motors connected to single-cylinder compressors 

Rolling mills 

Motors connected to mine hoists, elevators or cranes 



1.25 
1.33 
1.5 

1.67 

2 

3 to 4 

3 to 5 
6 to 8 

6 

4 

4 to 8 



Art. 284] 



NUT TALL COUPLING 



399 



(c) From Table 92, select the factor for the class of service for 
which the coupling is intended and multiply it by the horse 
power transmitted per 100 revolutions per minute. 

(d) Compare the horse power determined in (c) with the horse 
power rating of the coupling selected in (a) above. In case the 
latter is less than the former, select a larger coupling having the 
desired rating. 

(e) If the required speed is in excess of that listed for the cast- 
iron coupling, use a steel coupling. 

284. Nuttall Coupling.— The Nuttall coupling illustrated in 
Fig. 211 differs considerably from those discussed in the preced- 
ing articles, in that the power is transmitted through the medium 




Fig. 211. 



of helical springs c. These springs with the inserted case-hard 
ened plugs d are fitted into pockets between the twin-arms of 
the spider b. The casing a is provided with a series of lugs that 
fit loosely in the twin-arms of the spider and also bear against 
the spring plugs d. It is evident that with the construction 
shown in the figure this coupling can transmit power in either 
direction, and, furthermore, that the springs are always in com- 
pression. The clearance between the ends of the spring plugs is 
made slightly less than the maximum deflection of the spring; 
therefore, a sudden overload cannot break the springs. The 
coupling has a smooth exterior, hence there is not much, danger 
of injury to workmen. 



400 



CLARK COUPLING 



[Chap. XV 



285. Clark Coupling. — An interesting form of flexible coupling 
that was placed upon the market recently is the Clark coupling 
shown in Fig. 212. It consists of two hubs upon the flanges of 
which are cut a number of special teeth. Over these teeth is 
fitted a roller chain as shown in the figure. The teeth are cut 





accurately so that all of the rollers in the chain are in contact 
with the teeth, thus insuring an equal distribution of the load 
transmitted by the coupling. Side clearance is provided between 
the chain and the teeth, thus permitting the two halves of the 
coupling to take care of any slight angular displacement of the 




shafts. The chain is provided with a master link which may 
be removed quickly in case it is desired to run each shaft 
independently. 

286. Kerr Coupling. — -A type of flexible coupling particularly 
well adapted to very high rotative speeds is that shown in Fig. 
213. It was developed by Mr. C. V. Kerr for use in connecting 



Art. 286J 



KERR COUPLING 



401 



steam turbines to centrifugal pumps and blowers. In order to 
make it possible to use this coupling at high speeds, the dimen- 
sions are all kept down to a minimum by making the various 
parts of crucible cast steel. The through keys or cotters are 
made of tool steel and tempered. Due to the arrangement of the 
through keys at right angles to each other, the two shafts to be 
connected may be out of alignment to a considerable extent. To 
prevent serious wear of the various parts and to eliminate excess- 
ive noise, the coupling is filled with a heavy machine oil, or grease 
and graphite. To design a coupling of this kind the following 
method of procedure is suggested : 

(a) Design the shaft so that it will readily transmit the re- 
quired horse power at the specified speed. 




Fig. 214. 



(6) Design the cross-key so that it will be amply strong against 
failure due to crushing, shearing, and bending. 

(c) Design -the shell so that it will transmit the torsional mo- 
ment of the shaft. The key-ways in the shell should be investi- 
gated for crushing. 

287. Rolling-mill Coupling. — Frequently, flexible couplings are 
required in places where considerable grit, water, steam, etc., are 
present, and where noise is not objectionable; for example, in 
a rolling mill. For such and other heavy service, the rolling- 
mill type of flexible coupling shown in Fig. 214 is recommended. 
When the load transmitted is practically constant, a rolling-mill 
coupling will not be excessively noisy and good results may be 
expected. 



402 



JAW CLUTCH 



[Chap. XV 



RELEASING COUPLINGS 

A releasing coupling, or clutch, as it is commonly called, is 
so constructed that the connected shafts may be disengaged at 
will. From this statement it should not be inferred that clutches 
are used for connecting shafts exclusively, as they are also used 
for engaging pulleys, gears and other rotating parts. Clutches 
may be divided into two classes namely: (a) Positive clutches; 
(b) friction clutches. The latter class will not be discussed in this 
chapter, but will be taken up in detail in the following chapter. 

288. Positive Clutch. — The simplest form of positive clutch 
is the jaw clutch shown in Fig. 215(a). One part of the clutch 
is keyed or pinned rigidly to the shaft while the other part is 
splined, thus permitting it to be engaged with, or disengaged 
from, the first part by sliding it along the shaft. The interlock- 
ing jaws upon the abutting faces of the clutch may have various 




(d) 



(€) 



forms, as shown in Fig. 215. The jaws of the type shown in 
(6) engage and disengage more freely than square jaws. The 
jaws illustrated by (c), (d), and (e) are intended for installations 
where it is necessary to transmit power in only one direction. 
In punching and shearing machines the types of jaws shown 
by (a) and (e) are used considerably. 

289. Analysis of Jaw Clutches. — Having decided upon the 
type of clutch to be used for a particular installation, the next 
step calls for the determination of the dimensions of the several 
parts. In general, jaw clutches are designed by empirical rules, 
and consequently the resultant proportions are liberal. How- 
ever, if it is desired to arrive at the proportions of a jaw clutch 
capable of transmitting a certain amount of power, the following 
analysis is suggested: 

(a) Bore of the sleeves. — The bore of the sleeves is fixed by the 
size of the shaft required to transmit the required power. 



Art. 289] JAW CLUTCH ANALYSIS 403 

(b) Length of the sleeves. — If keys are used for fastening the 
sleeves to the shafts, the lengths of the sleeves are fixed, in a 
general way, by the length of keys required to transmit the de- 
sired power. In connection with punching and shearing 
machinery, where the clutch sleeve is occasionally fitted onto a 
squared shaft, the length of the sleeve may be assumed approxi- 
mately equal to the diameter of the shaft. 

(c) Outside diameter of sleeves. — The outside diameter of the 
sleeves must be such that the safe shearing strength of the jaws 
will exceed the pressure coming upon them. The pressure upon 
the jaws should be calculated on the assumption that it is con- 
centrated at the mean radius of the jaws. 

Let A = area of the jaw at the root. 

D = outside diameter of the clutch sleeve. 
S s = permissible shearing stress of the material. 
T = torsional moment to be transmitted by the clutch. 
d = bore of the clutch sleeve. 
n = number of jaws on the clutch sleeve. 

Equating the torsional moment T, to the moment of the 
shearing resistance of the jaws, and solving for the total re- 
quired shearing area, we obtain the following expression: 

nA - XDTdjs: (416) 

Without introducing any appreciable error, the area nA may be 
taken as equivalent to one-half the area between the circles having 
diameters equal to the outer and inner diameters of the clutch 
sleeve. Substituting for nA an expression for the equivalent 
area in terms of D and d, we arrive at the following relation: 

(Z)» - d*)(D + d) = ^f (417) 

In determining the outer diameter D by the use of (417), con- 
siderable time may be saved by solving this equation by trial. 

(d) Number and height of jaws. — The number of jaws on 
clutches depends upon the promptness with which a clutch must 
act. In punching and shearing machinery, the number of jaws 
varies from two to four, while in other classes of machinery the 
number of jaws may run as high as twenty-four. 

The height of the jaws must be such that the pressure coming 
upon them does not exceed the safe crushing strength of the 



404 JAW CLUTCH ANALYSIS [Chap. XV 

material used in the clutch. The distribution of the pressure 
upon the face of the jaws depends upon the grade of workman- 
ship put upon the clutch parts. On clutches found on the 
modern machine tools, we may safely say that the workmanship 
is of such a quality that the pressure upon the jaws may be 
assumed as uniformly distributed. 

Denoting the area of the engaging face of one jaw by the 
symbol A c , and the permissible crushing stress of the material 
by S c , we obtain the following relation by equating the torsional 
moment T to the moment of the resistance to crushing: 

A ° - m^fm (418) 

Having determined the area required to prevent crushing of the 
jaw, the height h of the latter is given by the following expression: 

A c 



D 



(419) 



Frequently, the height of the jaw as determined by (419) is so 
small that it must be increased in order that the mating jaws 
will hook together sufficiently and not be disengaged by any 
jarring action. Good judgment should play an important part 
in arriving at the various dimensions of the parts of a jaw clutch. 

References 

Elements of Machine Design, by W. C. Unwin. 

Machine Design, Construction, and Drawing, by H. J. Spooner. 

Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 

Design of Punch and Shear Clutches, Am. Mach., vol. 36, p. 991. 

Mechanical Engineers' Handbook, by L. S. Marks, Editor in Chief. 

Bulletin No. 4818A-Couplings, by General Electric Co. 

The Universal Joint, Am. Mach., vol. 38, p. 108. 

Friction Losses in the Universal Joint, Trans. A. S. M. E., vol. 36, p. 461. 

Catalogs of Manufacturers. 



CHAPTER XVI 
FRICTION CLUTCHES 

290. Requirements of a Friction Clutch. — The object of 
a friction clutch is to connect a rotating member to one 
that is stationary, to bring it up to speed, and to transmit 
the required power with a minimum amount of slippage. In 
connection with machine tools, a friction clutch introduces what 
might be termed a safety device in that it will slip when the pres- 
sure on the cutting tool becomes excessive, thus preventing the 
breakage of gears or other parts. 

In designing a friction clutch, the following points must be 
given careful consideration: 

(a) The materials forming the contact surfaces must be 
selected with care. 

(b) Sufficient gripping power must be provided so that the 
load may be transmitted quickly. 

(c) In order to keep the inertia as low as possible, a clutch 
should not be made too heavy. This is very important in high- 
speed service, such as is found in motor cars. 

(d) Provision for taking up wear should be made. 

(e) Provision should be made for carrying away the heat that 
is generated at the contact surfaces. 

(/) A clutch should be simple in design and contain as few parts 
as possible. 

(g) The construction should be such as to facilitate repair. 

(h) The motion should be transmitted without shock. 

(i) A clutch should disengage quickly and not "drag. " 

(j) A clutch transmitting power should be so arranged that no 
external force is necessary to hold the contact surfaces together. 

(k) A clutch intended for high-speed service must be balanced 
carefully. 

(I) A clutch should have as few projecting parts as possible, 
and such parts as do project should be covered or guarded so that 
workmen cannot come into contact with them. 

405 



406 CLUTCH FRICTION MATERIALS [Chap. XVI 

291. Materials for Contact Surfaces. — In order that a material 
may give satisfactory service as a frictional surface, it must fulfill 
the following conditions : (1) The material must have a high coef- 
ficient of friction. (2) The material must be capable of resisting 
wear. (3) The material must be capable of resisting high tem- 
peratures, caused by excessive slippage due to frequent operation 
of the clutch. 

Among the materials met with in modern clutches are the 
following : 

(a) Wood. — In many clutches used on hoisting machinery, as 
well as in some used for general transmission purposes on line- and 
counter-shafts, the contact surfaces are made of wood and cast 
iron. Among the kinds of wood that have proven satisfactory 
in actual service are basswood, maple, and elm. 

(b) Leather. — The majority of the cone clutches in use on 
motor cars are faced with leather. Some manufacturers use 
oak-tanned, while others prefer the so-called chrome leather. To 
obtain the best service from a leather facing, it should be treated 
by soaking it in castor oil or neat's-foot oil, or boiling it in tallow. 
Before applying the facing to the clutch, the treated leather 
should be passed between rolls so as to remove the excess oil or 
grease. Leather facings should never be allowed to become dry 
or hard, or the clutch will engage too quickly. Leather that has 
become charred due to excessive slippage has very little value as 
a friction material. 

(c) Asbestos fabric. — At the present time there are upon the 
market several patented asbestos fabrics consisting mainly of 
asbestos fiber. To give it the necessary tenacity the asbestos 
fiber is woven onto brass or copper wires. Among the well- 
known asbestos fabrics used for clutches, as well as for brakes, 
are Raybestos, Thermoid, and Non-Burn. The first two may be 
obtained in thicknesses varying from }£ to }/± inch, inclusive, and 
in widths of 1 to 4 inches, inclusive. Non-Burn is made in thick- 
nesses up to and including 1 inch, and in widths up to and in- 
cluding 24 inches. Asbestos fabric facings are used to a limited 
extent on cone clutches, and on a large number of modern disc 
clutches. When it is used on the latter type of clutch, the fabric 
may be riveted to the driving or to the driven discs, whichever is 
the more economical. 

The Johns-Manville Co. manufactures an asbestos-metallic 
block that is giving excellent service on clutches and brakes. 



Art. 291] CLUTCH FRICTION MATERIALS 407 

The block is constructed of long-fiber asbestos, reinforced with 
brass wire and moulded under an enormous pressure into any- 
desired shape. 

The main advantages claimed for the use of wire-woven as- 
bestos fabric and asbestos-metallic block are the following : 

1. Slightly higher coefficient of friction. 

2. Ability to withstand high temperatures. 

3. May be run dry or with oil. 

4. Not affected by moisture. 

5. Ability to resist wear. 

(d) Paper. — Compressed strawboard may be used as a fric- 
tion surface on clutches in which the speeds and the pressures 
coming upon the contact surfaces are low. If excessive slippage 
occurs, the strawboard is liable to become charred rather rapidly. 
Vulcanized fiber, which is nothing more than a form of paper 
treated chemically, gives fairly good service as a friction material 
in clutches. It is capable of withstanding medium pressure, as 
well as considerable slippage. 

(e) Cork inserts. — Cork is never used alone as a friction ma- 
terial, but always in connection with some other material either 
of a fibrous or a metallic nature. It is frequently used on leather- 
faced cone and metallic disc clutches, and is generally in the form 
of round plugs or inserts. The surface covered by these cork 
inserts varies from 10 to 40 per cent, of the total frictional area. 
Due to the higher coefficient of friction of cork, a motor-car 
clutch equipped with cork inserts is capable of transmitting a 
little more power for the same spring pressure than a similar 
clutch lined with leather; or for the same power, the spring pres- 
sure in the former is less than in the latter type of clutch. Cork 
inserts are also used on hoisting-drum cone clutches having wood 
blocks, and on common transmission clutches of the disc type. 
Experience has shown that they give excellent service. In gen- 
eral, the cork inserts are operative only at low pressures, as in 
engaging the clutch. In combination with the cork, the metal, 
leather, or wood in which it is imbedded forms the surface in con- 
tact after full engagement. Cork inserts also aid in keeping the 
surfaces lubricated. 

(/) Metallic surfaces. — The materials discussed above are all of 
a fibrous nature, and are always used in conjunction with a 
metal, such as cast iron, steel casting, steel, or bronze. Fre- 
quently, cone clutches used 'on machine tools have both cones 



408 CONE CLUTCHES [Chap. XVI 

made of cast iron, while in other cases cast iron and steel casting 
are used. Disc clutches using hard saw-steel discs running in oil 
are advocated by some manufacturers; others use steel against 
bronze, cast iron against bronze, and cast iron against cast iron. 
In all of the clutches using the metal-to-metal surfaces, a liberal 
supply of oil is furnished by some means or other. 

292. Classification of Friction Clutches. — According to the 
direction in which the pressure between the contact surfaces 
is applied, friction clutches may be divided into two general 
classes, as follows : 

(a) Axial clutches, which include all those having the contact 
pressure applied in a direction parallel to the axis of rotation. 
This class includes all types of cone and disc clutches. 

(b) Rim clutches, which include all clutches having the con- 
tact pressure applied upon a rim or sheave in a direction at right 
angles to the axis of rotation. 

AXIAL CLUTCHES 

A study of the designs of the clutches manufactured by the 
various builders of transmission machinery, machine tools, and 
motor cars, shows that axial clutches are made in a variety of 
forms. Such a study leads to the following classification of 
axial clutches: (1) cone; (2) disc; (3) combined conical disc. 

CONE CLUTCHES 

The cone clutch is without doubt the simplest form of friction 
clutch that can be devised, and if properly designed will give 
entire satisfaction. Two types of cone clutches are commonly 
met with, as follows: (1) single-cone; (2) double-cone. 

293. Single-cone Clutch. — The elements of a simple cone 
clutch are shown clearly in Fig. 216. The clutch consists of a 
cone (b) keyed rigidly to the shaft a, while a second cone d is 
fitted to the shaft c by means of the feather key e. This key 
permits the cone d to be engaged with the cone b, thus trans- 
mitting the power from one shaft to the other. The hub of the 
cone d is fitted with a groove /, into which is fitted the shifter 
collar operated by the engaging lever. 

(a) Machine-tool cone clutch.— A good example of the use of a 
simple cone clutch, applied to a machine tool, is shown in Fig. 



Art. 293] 



CONE CLUTCHES 



409 




Fig. 216. 




Fig. 217. 



410 CONE CLUTCHES [Chap. XVI 

217. The design shown is that used on the main driving pulley 
of the Lucas boring machine. The driving pulley a runs loose 
on the hub of the main bearing c, and has bolted to it the cast- 
iron cone b. The sliding cone d is fitted into 6 and is keyed to 
the main driving shaft A; by a feather key. The cones are engaged 
by means of the sliding spool h and the levers e and /. Several 
helical springs, one of which is shown in Fig. 217, are placed into 
holes drilled into the hub of the cone d; the function of these 
springs is to disengage the two cones when the spool h is moved 
toward the end of the sleeve g. It is quite evident that this 
clutch fulfills the important requirement met with in machine 
tools, namely, compactness and simplicity of design and ease of 
operation. 

(6) Motor-car cone clutch. — With the development of the 
modern automobile, the design of cone clutches was given more 
attention, and at the present time approximately 40 per cent, 
of the pleasure-car manufacturers are equipping their cars with 
clutches of the single-cone type. In motor cars the clutch is 
used to connect the motor to the transmission, and normally 
is held in engagement by a spring pressure. This spring pressure 
must be released by the pedal when the car is to be stopped or 
when speed changes are made by shifting the gears. 

National clutch. — In Fig. 218 is shown a design of a cone 
clutch used on the National motor car. The cone a with its 
various attachments is forced into the conical bore of the fly- 
wheel rim by the pressure of the helical spring. To decrease the 
weight of the clutch, the cone a is made of aluminum having its 
periphery faced with leather. The small flat springs b, with 
which the cone is fitted at various points along its periphery, 
provide the smooth and easy engagement so desirable in motor 
cars. To prevent spinning, the sliding sleeve c has fastened to 
it a small brake sheave d upon which a brake block e acts. The 
brake block is fitted to an operating lever / which is depressed 
when the clutch is disengaged. 

Cadillac clutch. — The clutch shown in Fig. 219 is that used 
on the old four-cylinder Cadillac motor car. It differs consider- 
ably from the National clutch discussed above. The cone a is 
made of pressed steel, and the flywheel instead of having its rim 
bored conical has a special rim b fastened to it. The pressure 
forcing the cone a into the cone b is produced by a series of springs 
in place of a single central spring as in the preceding case. A 



Art. 293] 



CONE CLUTCHES 



411 



possible advantage of this arrangement is that the adjustment 
for wear may be made more easily; also the pressure may be 
distributed more uniformly over the surface in contact. 




Fig. 218. 



294. Double-cone Clutch. — The clutches described in Art. 
293 were all of the single-cone type. In connection with hoist- 



412 



CONE CLUTCHES 



[Chap. XVI 



ing machinery, machine tools, and motor cars, it is not unusual 
to find double-cone clutches. 

(a) Clyde clutch. — The design of a double-cone clutch used on 
hoisting drums manufactured by the Clyde Iron Works of 




Duluth, Minn., is shown in detail in Fig. 223. The friction 
blocks c forming one member of the clutch are fastened to the 
gear b, which is keyed to the shaft a. The clutch is engaged by 
moving the drum d along the shaft a by means of the combination of 



Art. 295] 



CONE CLUTCH ANALYSIS 



413 



lever k, screw h, thrust pin g, cross-key /, and collar e. A spring 
I, located between the drum and the gear, automatically disen- 
gages the clutch when the thrust on the cross-key is released. 

In place of using a double-cone clutch, several manufacturers 
of hoisting drums employ one of the single-cone type, operated 
practically in the same manner as explained in the preceding 
paragraph. The Ingersoll slip gear, described in Art. 231, is 
nothing more than a form of double-cone clutch. 

295. Force Analysis of a Single -cone Clutch. — In the follow- 
ing analysis of a single-cone clutch, we shall assume that the 
outer cone is the driving member while the inner cone is the 
member having an axial motion. In Fig. 
220 are shown the various forces acting 
upon the inner cone. It is required to 
determine an expression for the moment 
M that the clutch is capable of trans- 
mitting for any magnitude of the axial 
force P. 
Let p = the unit normal pressure at 

the surface in contact. 

r x = the minimum radius of the 

cone. 

r 2 = the maximum radius of the 

cone, 
ju = the coefficient of friction. 

The maximum moment that the clutch 
will transmit is equivalent to the moment 
of the frictional resistance between the 
inner and outer cones. The normal force acting upon an elemen- 
tary strip of the surface in contact is 2irrp —. — . The com- 

ponent of this normal pressure parallel to the line of action of 
the axial force P is 2 rrpdr. The summation of these compo- 
nents over the entire surface in contact must equal P; hence 

P = 2wfprdr (420) 

The force of friction upon the elementary strip is 2ir^rp — 

dr 

and its moment about the axis is 2 irur 2 p —. . Therefore the 

^ l sina 

moment M is given by 




Fig. 220. 



414 CONE CLUTCH ANALYSIS [Chap. XVI 

M = i^E [prHr (421) 

sin a J r v ' 

With our present knowledge of friction, it is impossible to 
determine a correct expression for the moment of the frictional 
resistance between the two elements of a cone clutch. From (421) 
it is evident that an expression for the moment of friction de- 
pends upon the distribution of the pressure between the contact 
surfaces as well as the variation of the coefficient of friction. 
When the clutch is new and the surfaces are machined and fitted 
correctly, it is probable that the pressure is nearly uniformly 
distributed. However, after the clutch has been in service for 
a period of time, there will be a redistribution of the pressure 
due to the unequal wear caused by the different velocities along 
the surfaces in contact. This variation in velocity no doubt 
results in a change in the value of the coefficient of friction. In 
view of the fact that no experimental data are available, we shall 
assume that the coefficient of friction remains constant, and 
further, that the normal wear at any point is proportional to the 
work of friction. Denoting the normal wear at any point by 
n, the law just stated may be expressed by the relation 

n = kpr ' (422) 

Assuming that the surfaces in contact remain conical, it follows 
that the normal wear is constant; hence 

V = ~ r > (423) 

in which C denotes the ratio of the constants n and k. Substi- 
tuting the value of p from (423) in (420) and (421), and integrat- 
ing between the proper limits, we obtain the following relations : 

P = 2tC(t 2 - n) (424) 

M = ^- (r 2 2 - r\) (425) 

Sin a v 2 l 

Eliminating C between (424) and (425) , we get finally 

M = J™-, (426) 

2 sin a 

in which D represents the mean diameter A B of the cone shown 
in Fig. 220. 



Art. 295] CONE CLUTCH ANALYSIS 415 

To determine the horse power that a cone clutch will transmit, 
substitute the value of M from (426) in the formula 

H - 6P50 (427) 

whence 

H = I26^60"sina' (428) 

and the axial force is 

P = ^.OMfls ina (42g) 

The total normal pressure is given by the following expression : 



2irC C r l 2tC 

■r n = -■ I dr = 

sin aj ri y sm a 



(r 2 - ri) 



Eliminating C by means of (424), it is evident that 

P n = -£t (430) 

sm a 

The total normal pressure is also equal to the average intensity of 
unit normal pressure multiplied by the total area in contact; or 

S~ = v Dfp' (431) 

sin a J ^ 

Combining (428) and (431), and solving for H, we get 

Denoting the product of /j. and p' by the symbol K, (432) becomes 

_ KfND* 
H - ~4A%m> (433) 

By means of (433) it is possible to determine values of the de- 
sign constant K for cone clutches in actual service. Such values, 
if based on clutches in successful operation, will prove of consider- 
able help in the design of new clutches. The analysis used in 
deriving (433) is similar to that first proposed by Mr. John Edgar 
in the American Machinist of June 29, 1905, though he applied 
his formulas to expanding ring clutches. 

Another design constant that may be found useful in arriving 
at the proportions of a cone clutch is that which represents the 
number of foot-pounds of energy per minute that can be trans- 



416 STUDY OF CONE CLUTCHES [Chap. XVI 

mitted per square inch of contact surface of the clutch. Denot- 
ing this constant by the symbol K h we find that 

K , . m°M (434) 

296. A Study of Cone Clutches. — Through the generosity and 
cooperation of about forty automobile manufacturers, informa- 
tion pertaining to a large number of cone clutches was obtained. 
Some of the clutches that were analyzed were faced with leather, 
others with asbestos fabric, and a few were equipped with cork 
inserts. 

From the information furnished by the various manufacturers, 
it was possible to determine for each clutch the magnitude of the 
design constant K and the intensity of the unit normal pressure 
p'. With K and p' known, the probable value of the coefficient 
of friction /j, was calculated. The values of K, p r , and n were 
found to vary with the mean velocity of the surface in contact. 
In this, as well as in all other analyses of motor-car clutches, the 
values of K are based upon the horse power and speed correspond- 
ing to the maximum torque of the motor, and not upon the maxi- 
mum horse power transmitted and the speed corresponding 
thereto. It should be remembered that clutches must be de- 
signed for the maximum loads coming upon them, and in the 
case of motor cars, the loads are greatest when the motor trans- 
mits the maximum torque. 

(a) Leather-faced cone clutches. — For the leather-faced cone 
clutches analyzed, the values of K and p' were plotted on a speed 
base, and the curves shown in Fig. 221 represent the average 
results. From this figure, it is apparent that the magnitude of 
K decreases with an increase in the mean velocity of the sur- 
faces in contact. The intensity of the unit normal pressure p' 
also decreases with an increase in the velocity. The value of the 
coefficient of friction was also plotted on a speed base, and an 
average curve passed through the series of points. For all prac- 
tical purposes, the average value of /j, may be represented by a 
straight line parallel to the velocity axis, giving a constant value 
of n equal to 0.2 for all speeds. The p curve is not shown in 
Fig. 221. 

(6) Cone clutches faced with asbestos fabric. — At the present 
time there are only a few motor-car builders using cone clutches 
faced with asbestos fabric. From an investigation of six such 



Art. 296] 



STUDY OF CONE CLUTCHES 



417 



clutches, K was found to vary from 1.95 to' 4.77. The intensity 
of the unit normal pressure p' varied from 9.5 to 17 pounds per 
square inch. Until such a time as more information pertaining 
to asbestos fabric facing is available, it is suggested that the 
values of K given in Fig. 221 be used, and that the coefficient of 
friction be assumed as 0.30. 

(c) Cone clutches with cork inserts. — It was impossible to get 
information pertaining to a large number of cone clutches having 
cork inserts, since very few motor-car builders are using them at 
the present time. Four such clutches were analyzed and the 



4 

c 

£ 3 

m 

c 

O 

c 

S 2 

c 

m 
c 

> 




K 


































































20°- 
o 

10 1 














S 
















s 










N 










\ 














































































S 


















N 






















V 


































































































































































=^ 






































































































































































































































































































































• 























































































































































































































































3000 3500 4000 

Velocity in ft. per min. 
Fig. 221. 



design constant K was found to vary from 2.2 to 3.1. Until such 
a time as sufficient information is available for a more extended 
analysis, it would seem advisable to use the values of K given 
for the leather-faced cones when making calculations for cork 
insert clutches. The coefficient of friction may be assumed as 
0.25. 

(d) Cone-face angle. — In the study of the motor-car cone 
clutches, it was found that for a leather facing the face angle a 
varied from 10 to 13 degrees. The majority of the manufac- 
turers are using 12^ degrees which is now recommended as a- stand- 
ard by the Society of Automotive Engineers. With an asbestos- 



418 



CONE CLUTCH INVESTIGATION 



[Chap. XVI 



fabric facing, the angle a varied from 11 to 14)^ degrees, and for 
the cork insert clutches, from 8 to 12 degrees. 

297. Experimental Investigation of a Cone Clutch. — In the 
Zeitschrift des Vereines deutscher Ingenieure, for Dec. 15, 1915, 
Prof. H. Bonte of Karlsruhe presented an article in which he gave 
the results of an experimental investigation of a cone clutch. 
The two halves of the clutch were made of cast iron, and during 



600 
500 

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C 

T3 400 
o; 

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4- 

E 

c 300 
c 

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V 






























































































































v 






























































/ 



























































































































































































































































































































































































































































































































































































































































































































































































































































































































50 



100 


150 200 


ui al 


Pre s sure 




Fig. 222. 



250 



300 



350 



lb. 



the test the surfaces in contact were lubricated. The main ob- 
ject in undertaking the investigation was to determine which of 
the following two formulas, generally quoted in technical works, 
was the correct one to use in designing cone clutches : 

£2- '(435) 

2 sin a v 



M = 



M = 



txPD 



2 (sin a + fx cos a) 



(436) 



Art. 297] CONE CLUTCH INVESTIGATION 419 

In Fig. 222 are plotted the results of Prof. Bonte's experimental 
investigation on a clutch having an angle a = 15 degrees. In 
this figure are included the results obtained by evaluating (435) 
and (436). The results obtained by the use of (435) are rep- 
resented by the dot and dash line, and those obtained by the 
use of (436) are represented by the dash line. The following 
conclusions may be arrived at from the results published by 
Bonte. 

(a) When the angle a is 15 degrees, the error introduced by 
using (436) is large, while the agreement between the experi- 
mental results and those obtained by using (435) is very close. 

(6) When the angle a is 30 degrees, the experimental results 
lie between those obtained by (435) and (436). The curve repre- 
senting the results obtained by (435) lies above, but is much 
closer to the experimental curve than that obtained by (436). 

(c) For the angle 45 degrees and 60 degrees, the experimental 
points lay above those obtained by (435), which in turn lay 
above the points obtained by (436). 

(d) Apparently, the coefficient of friction is not constant as 
generally assumed but varies slightly with the pressure. 

As a result of this experimental investigation, Prof. Bonte 
makes a plea that (436), which is apparently incorrect, should 
no longer be used in designing cone clutches. 

298. Analysis of a Double-cone Clutch. — For the double-cone 
clutch shown in Fig. 223, it is required to determine an expres- 
sion for the force F that must be applied at the end of the lever 
k in order to engage the clutch; also, to determine the maximum 
moment that the clutch is capable of transmitting. 
Let Z>i = the mean diameter of the smaller cone. 
D 2 = the mean diameter of the larger cone. 
D 3 = the mean diameter of the thrust collar e. 
Z>4 = the mean diameter of the spring cage m. 
L = the length of the lever arm k. 

P = the axial force holding the drum against the V blocks. 
S = the spring force. 
d = the mean diameter of the screw h. 
P = the angle of rise of the mean helix of the screw. 
<p' = the angle of friction for the screw, 
/no = the coefficient of friction between the drum and 

blocks c. 
ix = the coefficient of friction between the drum and 
collar e and cage m. 



420 



DOUBLE-CONE CLUTCH ANALYSIS [Chap. XVI 



Consulting Fig. 223, it is evident that the axial pressure that 
the screw h must produce is P + S plus the force required to 
move the drum with its load along the shaft. The latter force is 
relatively small and may be accounted for by considering it 
equivalent to a certain percentage of the total pressure produced. 
Calling Q the total pressure due to the screw, we find that its 
magnitude may be expressed by the formula 

Q = i — L ^' (437) 

v 

in which rj may be assumed as equal to 0.97. From this it 
follows that the force F required on the operating lever h, in 




Fig. 223. 



order to produce the axial pressure Q, must have a magnitude 
given by the expression 



F=f£tanG8 + *') 



(438) 



The total moment that the drum will transmit is equivalent 
to the sum of the moments of friction of the double cone c, of 
the thrust collar e, and of the spring cage m. The last two 
moments just mentioned are usually small when compared with 
the first, and frequently are not considered at all. The moment 



Art. 298J 



DOUBLE-CONE CLUTCH ANALYSIS 



421 



transmitted by the double cone is equivalent to the sum of the 
moments of the two cones taken separately, or 

Mi + M 2 = j^f (Di + D 2 ) (439) 

4 Sin o: 

The sum of the moments transmitted by the collar e and the 
spring cage m is 

Ms + Mi = ^ (P + S) + ^ (440) 

Adding (439) and (440), we find that the total moment trans- 
mitted by the drum has the magnitude 

M = M x + M 2 + M s + Mi (441) 

299. Smoothness of Engagement of Cone Clutches. — In 

motor-car service, it is very desirable that the car be started 




without jerks. In order to secure smooth clutch engagement, 
the designers of clutches were compelled to originate devices 
that insured evenness of contact between the friction surfaces. 
A few such devices, as applied to cone clutches, are shown in 
Figs. 224 to 227, inclusive. In general, it may be said that the 
function of these devices is to raise slightly the cone facing 
at intervals around the periphery, so that upon engagement 
only a small portion of the friction surface comes into contact 
with the flywheel rim. As soon as the full spring pressure 
is exerted, the facing is depressed and the entire surface of 
the cone becomes effective. One disadvantage of the attach- 



422 



CONE CLUTCH ENGAGING DEVICE [Chap. XVI 



ments just discussed is that they tend to increase the spinning 
effect due to the extra weight added to the periphery of the cone. 




Fig. 226. 





(a) 



Fig. 227. 



(b) 



A few manufacturers are using cork inserts in connection 
with their leather-faced cone clutches. It is claimed that in 



Art. 301] SINGLE DISC CLUTCHES 423 

addition to increasing the coefficient of friction between the 
surfaces in contact, the cork inserts have the effect of producing 
smooth and easy engagement of the clutch. Obviously, cork 
inserts have another advantage in that the weight of the cone 
is actually decreased, thereby decreasing the spinning effect. 
Fig. 227(6) shows one method of holding cork inserts in the facing 
of a cone clutch. 

300. Clutch Brakes. — In addition to securing smooth and easy 
clutch engagement, some means must be provided to prevent 
the "spinning" of the clutch when it is disengaged. By keeping 
the size and weight of the clutch down to a minimum, spinning 
may be reduced slightly. However, to overcome the spinning 
action completely, small brakes that are brought into action when 
the pedal is depressed must be provided. A cone clutch equipped 
with such an auxiliary brake is shown in Fig. 218, and in Figs. 
229, 236, and 241 are shown disc clutches equipped with such 
brakes. 

DISC CLUTCHES 

In general, a disc clutch consists of a series of discs arranged 
in such a manner that each driven disc is located between two 
driving discs. Disc clutches are made in various forms, as a 
study of the designs used in connection with various classes of 
machinery will show. For convenience, disc clutches will be 
classified as follows: 

(a) Single-disc type, in which a single disc serves as the driven 
member. 

(6) Multiple-disc type, in which two or more discs act as the 
driven member. 

301. Single-disc Clutch.— In Figs. 228 to 233, inclusive, are 
shown six designs of single-disc clutches, the first two represent- 
ing the practice of two motor-car builders, and the third and 
fourth showing the details of two clutches used for general power- 
transmission purposes. The remaining two, namely, those 
shown in Figs. 232 and 233, are intended for special purposes. 
As in the case of the cone clutches, the development of the auto- 
mobile is responsible to a large extent for the advances made in 
the design of disc clutches. 

(a) Knox clutch. — The disc clutch shown in Fig. 228 is that 
used on the old Knox motor cars. The discs a and b are fastened 



424 



KNOX CLUTCH 



[Chap. XVI 



to the flywheel while the driven disc c is fastened to the flange d, 
which in turn is splined to the transmission shaft e. Due to the 
action of a series of springs located in the rim of the flywheel, the 
driven disc c is clamped between the two driving discs. The 
clutch is released by overcoming the spring force upon the discs 
b, through the medium of the sliding sleeve/, lever g, and plunger 




Fig. 228. 



h. All of the discs used in this clutch are made of cast iron. In 
order to obtain smooth engagement and to increase the coefficient 
of friction between the surfaces in contact, the driven disc c is 
fitted with cork inserts as shown. 

(6) Velie clutch. — The type of single-disc clutch used on the 
Velie motor cars is shown in Fig. 229. Instead of having two 



Art. 301] 



YELIE CLUTCH 



425 



driving discs as in the Knox clutch, this design has only one 
driving disc b, but the web of the flywheel serves the same pur- 
pose as a second disc. The. steel driven disc c is riveted to the 
flange of the clutch drive shaft d. The clutch is kept in engage- 
ment by the conical spiral spring pressing upon a bronze sleeve, 
which in turn transmits the pressure to the wedge / by means -of 




Fig. 229. 



suitable links. The back face of the driving disc b, as well as 
the inside face of the cover plate a, is bored conical to fit the 
wedge /. The cover plate screws into the flywheel and is locked 
to it by means of the set screws shown. To release the clutch, 
the wedge is withdrawn slightly by forcing the bronze sleeve 
back against the action of the spring. The treadle operates the 
releasing collar g by means of a system of links and levers. In 



426 



PLAMONDON CLUTCH 



[Chap. XVI 



the Velie clutch, the driven disc c is faced on both sides with an 
asbestos fabric, called Raybestos. Attention is directed to the 
small disc brake which prevents excessive spinning when the 
clutch is released. 

(c) Plamondon clutch. — A sectional view of the Plamondon disc 
clutch as applied to a pulley running loose on a shaft a, is shown 
in Fig. 230. The disc c, which is faced with hard maple seg- 
ments, is made in halves so that it can be removed in case the 
friction blocks require renewal. The flange d slides on the flanged 
hub e, which is keyed to the shaft a. By means of the compound 




toggle levers, /, g, and h, the flanges d and e are pressed against 
the disc c, thereby transmitting the power from the pulley to the 
shaft, or vice versa. Attention is called to the simplicity of this 
clutch and also to the ease with which adjustments for wear may 
be made. 

(d) E. G. I. clutch. — In Fig. 231 is shown another design of a 
single-disc clutch, but in this case the pressure upon the discs is 
produced by a system of rollers and levers instead of springs or 
toggle joints. The cast-iron discs c and d are made to rotate 



Art. 301] 



E. G. I. CLUTCH 



427 



with the casing b by means of the three bolts e. The casing b is 
fastened to the shaft a by set screws or keys. Between the slid- 
ing discs c and d is located a third disc I, to the hub of which may 
be fastened a gear or pulley. The pressure exerted by the sliding 
discs upon the disc I is produced by shifting the sleeve /inward. 
This movement causes the levers h to assume a position perpen- 
dicular to the shaft, thereby forcing apart the disc c and the 
casing b, and at the same time creating a considerable pressure 
upon the disc I. Upon disengagement of the clutch, the springs 




Fig. 231. 



m spread the discs c and d. The disc I is fitted with a series of 
wooden plugs, as shown in the figure. 

302. Hydraulically Operated Disc Clutch. — In such naval ves- 
sels as torpedo boat destroyers, it has been found that a combina- 
tion of reciprocating engines with turbines gives better economy 
over a wide range of speed than turbines alone. The engines are 
used for cruising speeds only, and exhaust into the low-pressure 
turbines. At the higher speeds, the ship is propelled by turbines 
only. According to the machinery specifications drawn up by 



428 



METTEN HYDRAULIC CLUTCH 



[Chap. XVI 



the Navy Department for some of the latest types of destroyers, 
the installation of turbines and cruising engines called for must 
fulfill the following conditions: 

(a) That the engines and turbines should be capable of oper- 
ating in combination on cruising speed. 

(6) That the turbines should be capable of operating alone, the 

engines standing idle. 

(c) That means should be 
provided whereby the cruising 
engines may be connected or 
disconnected to the turbine 
shafts without stopping the 
propelling machinery. 

It is evident from the above 
specifications that some form 
of reliable clutch is necessary 
to fulfill condition (c), and in 
order to meet this require- 
ment, Mr. J. F. Metten, Chief 
Engineer of the Wm. Cramp 
and Sons Ship and Engine 
Building Co., developed and 
patented the single disc clutch 
shown in Fig. 232. The hol- 
low crankshaft a of the re- 
ciprocating engine has con- 
nected to it the head b, which 
in combination with the steel 
frame c forms the driving 
member of the clutch. The 
inner face of the frame c is 
lined with an asbestos fabric. 
Inside of this driving member 
and attached to it, is located 
a movable member consisting of the spherical steel-plate head e, 
ring / faced with asbestos fabric, and the flexible ring g. The 
shaft I, which is an extension of the main turbine shaft, has bolted 
to its flange a steel-plate disc k, Y± inch thick. When oil under 
pressure is forced through the hollow crankshaft into the pressure 
chamber formed between the heads b and e and rings / and g, the 
disc k is gripped by the friction surfaces d and h. As shown in 




Fig. 232. 



Art. 302] METTEN HYDRAULIC CLUTCH 429 

Fig. 232, the clutch is disengaged. In order to insure quick dis- 
engagement of the clutch, the flexible ring g is so designed that 
its contraction upon release of the oil pressure will force the oil 
out of the pressure chamber. 

The axial force available for creating the frictional resistance 
on the disc k is that due to the fluid pressure upon the combined 
unbalanced areas of the head e and ring /, minus the resistance 
that the flexible ring g offers to extension. Having determined 
the axial force, and knowing the inner and outer diameters of the 
contact surfaces d and h, the probable horse power that the clutch 
is capable of transmitting may readily be determined. 

303. Slip Coupling. — In many installations, it is desirable to 
place between a motor and the driven machine or mechanism 
some form of coupling that will slip when the load is excessive, 
thus protecting the motor against overloads. The details of such 
a coupling, called a slip coupling, are shown in Fig. 233, which 
represents the design used by the Illinois Steel Co. and others 
on the drives of rolling mill tables. A modification of this design 
is also used on the furnace-charging machines found in steel 
works. The slip coupling illustrated in Fig. 233 is nothing more 
than a single-disc friction coupling. The flanged hub a is keyed 
to the driving shaft, and has bolted to its rim a plate b. Be- 
tween a and b, and separated from them by fiber discs, is the 
flanged hub c which is keyed to the driven shaft. The bolts con- 
necting the plate b with the hub a are provided with springs 
which create a pressure on both faces of the hub c. The torsional 
moment transmitted by the coupling depends directly upon this 
spring pressure, which may be varied by merely adjusting the 
nuts of the coupling bolts. In Table 93 are given the gen- 
eral proportions of a series of sizes of the slip coupling shown in 
Fig. 233, and these proportions represent the practice of the 
Illinois Steel Co. 

304. Multiple-disc Clutches.— In Figs. 234 to 237, inclusive, 
are shown four designs of multiple-disc clutches, the first two of 
which represent the practice of two manufacturers of transmis- 
sion clutches, and the last two show the type of multiple disc 
clutches used on motor cars. 

(a) Akron clutch. — The Akron clutch shown in Fig. 234 is a 
double-disc clutch employing an ingenious roller toggle for pro- 
ducing the pressure between the discs. The clutch consists of 



430 



SLIP COUPLING 



[Chap. XVI 



a casing a upon the hub of which gears, sprockets, or pulleys may- 
be keyed. Into the casing a is screwed a head b having a series 
of notches on its periphery, into which the locking set screw c 
projects. This combination of screwed head and set screws 
affords a simple and effective means of making adjustments for 
wear. The inner face of the head b and that of the casing a are 




machined and serve as contact surfaces for the discs d and e, 
respectively. The discs are splined to the hub /, which in turn 
is keyed to the shaft g. To engage the clutch, the sliding sleeve 
h is moved outward, thus pulling the forked levers k with it, 
and as a result of the action of the roller toggle, forcing the discs 



Art. 303] 



SLIP COUPLING 



431 







^ M 














































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432 



AKRON CLUTCH 



[Chap. XVI 



d and e apart. The clutch is lubricated effectively by having 
the casing partially filled with oil, and hence the wear on the 
friction surfaces is reduced to a minimum. 




Fig. 234. 



(6) Dodge clutch. — The multiple-disc clutch shown in Fig. 
235 is used for general transmission service. The cylindrical 
casing c with its hub h is keyed to the shaft a, and may serve 




either as the driving or the driven member. The discs e, fitted 
with wood blocks, rotate with c and at the same time may be 
moved in an axial direction. The flanged hub / is keyed to the 



Art. 3041 



ALCO CLUTCH 



433 



shaft k and has splined to it the two discs g and h, the outer one 
of which may be moved forward by the roller toggle operating 
mechanism. The axial movement given to h clamps the various 
discs together, thus transmitting the desired power. It should 
be noted that means for taking up wear on the discs are provided, 
and that the clutch is self-lubricating. An oil ring m revolves 




Fig. 236. 



upon the shaft and carries a continuous supply of lubricant 
from the oil reservoir below to all parts of the sleeve. 

(c) Alco clutch. — The multiple-disc clutch used on the Alco 
car, formerly manufactured by the American Locomotive Co., 
is shown in Fig. 236. As shown in the figure, the driving discs 
are connected to the flywheel through the hollow pin b and the 



434 



PATHFINDER CLUTCH 



[Chap. XVI 



drum a, while the driven discs are splined to the inner hub c 
which is keyed to the clutch shaft d. Both driving and driven 
discs are so mounted that they must rotate with the member to 
which they are connected, and at the same time these discs may 
move in an axial direction. To disengage the clutch, the collar 
e is moved to the right carrying with it the sleeve / and the spider 
g, thus releasing the pressure between the two sets of discs. As 
soon as the treadle is released, the spring will engage the clutch. 

Both sides of the driving 
discs are faced with 
Raybestos. 

(d) Pathfinder clutch. 
— Another form of multi- 
ple-disc clutch is shown 
in Fig. 237, and as in the 
Alco clutch, both sides 
of the driving discs are 
faced with asbestos 
fabric. The latter 
clutch is much shorter 
in length than the 
former, and in place of a 
single spring to create 
the axial pressure upon 
the discs, a double con- 
centric spring is used. 
The pressure upon the 
treadle is transmitted to 
the collar e on the trans- 
mission shaft d by the 
shipper arm k, through 
Fig. 237. the medium of a ball 

bearing m, as shown in 
the figure. In general, the description and operation of the Path- 
finder clutch is similar to that of the Alco clutch. 

305. Force Analysis of a Disc Clutch.— It is required to de- 
termine an expression for the moment M that the clutch is 
capable of transmitting for a given magnitude of the axial force 
P. We shall assume, in the following analysis, that the law 
expressed by (422) will hold for disc clutches. This is approxi- 
mately true, especially for clutches having very narrow contact 
surfaces. 




Art. 305] DISC CLUTCH ANALYSIS 435 

Let D = the mean diameter of the discs. 
r x = the minimum radius of the discs. 
r 2 = the maximum radius of the discs, 
s = the number of friction surfaces transmitting power. 

The general expressions deduced for the conical clutch may- 
be applied to the disc clutch by making the angle a = 90 de- 
grees. Substituting this value in (426), we get for the moment 
for each contact surface 

Hence for s surfaces, the total moment becomes 

M=^ (442) 

Substituting (442) in (427), we find that the horse power trans- 
mitted by a disc clutch is given by the expression 

from which the axial force is 

= 126,060 H 
r spDN { ^ V 

The total axial pressure P is also given by the product of the 
area of contact of one disc and the average intensity of normal 
pressure p', that is 

' P = T (r\-r\) p' (445) 

Combining (444) and (445), and solving for H, we obtain the 
following expression 

W'sDW, (446) 

in which / denotes the face of the contact surface, or (r 2 — n). 
Replacing up' by the symbol K 2 , as was done in Art. 295, (446) 
becomes 

H= *MPlKl (447 ) 

a 4012() \tti) 

In disc clutches, as in cone clutches, it may be desirable to 
know the number of foot-pounds of energy the clutch will trans- 



436 



STUDY OF DISC CLUTCHES 



[Chap. XVI 



mit per minute per square inch of actual contact surface, 
senting this factor by the symbol K 3 , we get 

10,500 H 



K, 



sfD 



Repre- 



(448) 



306. A Study of Disc Clutches. — (a) Motor-car dutches. — 
A study of disc clutches used on motor cars discloses the fact that 
the majority of such clutches have steel discs in contact with 
asbestos-fabric-faced steel discs. Among other combinations 
that are used for the friction surfaces, the following may be 



G 
> 


































































































































" h - 20 

O 

03 

> 


































































































































































































































































































































^ 0.2 
, <+- 
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ID 
CD 

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1000 



1500 



2000 



2500 



Mean Velocity - f+. per min. 
Fig. 238. 



mentioned: (1) steel against steel; (2) steel against steel with 
cork inserts; (3) steel against bronze. 

An analysis, similar to that of cone clutches, was made of a 
large number of different types of disc clutches used on motor 
cars. The information required for such an analysis was fur- 
nished by the various motor-car manufacturers. The graphs 
plotted in Fig. 238 represent the average results obtained for 
the asbestos-fabric-faced disc clutches running dry, and are 
based upon an investigation of at least thirty-five different 
clutches. The values of K 2 were obtained by evaluating (447)-, 



Art. 306] 



STUDY OF DISC CLUTCHES 



437 



while those of p' were deter- 
mined by means of (445). 
The graph for the coefficient of 
friction ju was established from 
the relation K 2 = up'. 

For clutches employing the 
other friction surfaces men- 
tioned in a preceding paragraph, 
it was thought best not to rep- 
resent the results graphically, 
since there was not sufficient in- 
formation available to warrant 
definite conclusions. However, 
in Table 94 are exhibited the 
minimum, maximum, and mean 
values of the design constant 
K 2 , of the average intensity of 
normal pressure p', and of the 
coefficient of friction for the 
various types of motor-car disc 
clutches investigated. 

(b) Transmission clutches. — 
Through the generosity of 
several manufacturers of trans- 
mission clutches, considerable 
information was obtained which 
made it possible to carry out an 
analysis similar to that on 
motor-car clutches mentioned 
above. Since no information 
regarding the axial pressure 
upon the discs was available, it 
was impossible to determine the 
probable values of p' and n, and 
consequently only the relation 
between the design constant K 2 
and the mean velocity of the 
friction surfaces at 100 revolu- 
tions per minute of the clutch 
was calculated. The reason 
for selecting the mean velocity 





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438 



STUDY OF DISC CLUTCHES 



[Chap. XVI 



at 100 revolutions per minute of the clutch as one of the vari- 
ables is the fact that all of the manufacturers rate their clutches 
at this speed. 

The disc clutches investigated were fitted with the following 
combinations of friction surfaces: cast iron against wood; cast 
iron against compressed paper and wood; cast iron against cast 
iron; cast iron against cast iron with cork inserts. 

1. For clutches equipped with cast-iron discs in contact with 
wood-faced discs, it was found that the design constant K.% 



25 

4- 

° 15 

in 
<u 

o 10 
> 

5 




\ 


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400 



100 200 300 

Mean Veloci+y - f+. per min 
Fig. 239. 



varied between wide limits. This variation is clearly shown in 
Fig. 239, in which the two curves represent the maximum and 
minimum results obtained. 

2. For clutches having cast-iron discs in contact with discs 
faced with compressed paper, the relation existing between K 2 
and the mean velocity at 100 revolutions per minute of the 
clutch is represented by the graph of Fig. 240. 

3. For clutches having cast-iron friction surfaces, the relation 
between the design constant i£ 2 and the mean velocity may be 
expressed by the following formula: 

50' 



K 2 



18 



(449) 



Art 307] 



HELE-SHAW CLUTCH 



439 



in which V denotes the mean velocity of the friction surfaces 
at 100 revolutions per minute of the clutch. 

4. For clutches having cast-iron discs in contact with cast- 
iron discs fitted with cork inserts, the relation between K 2 and 
the mean velocity of the friction surfaces is given by the fol- 
lowing expression: 

V 



K 2 = 17 



150 



(450) 



COMBINED CONICAL-DISC CLUTCHES 



By a combined conical-disc clutch is meant one in which the 
contact surfaces of the disc or discs are conical. Several de- 
signs of conical-disc clutches are available, the most important 
of which are described briefly in the following paragraphs. 




Mean Velocity - fh per mi 
Fig. 240. 



307. Hele-Shaw Clutch. — A sectional view of the Hele-Shaw 
multiple conical-disc clutch as used on motor cars is shown in 
Fig. 241. The driving and driven discs have a V-shaped annular 
groove, the sides of which form the surfaces in contact. The 
phosphor-bronze driving discs are provided with notches on the 
outer periphery which engage with suitable projections b on the 
pressed steel clutch casing a. The mild steel driven discs have 
notches on the inner bore which engage with the corresponding 
projections on the steel spider c. This spider is splined to the 
driving shaft, as shown in Fig. 241. The V groove in the discs 
permits a free circulation of oil, and at the same time insures 
fairly rapid dissipation of the heat generated when the clutch is 
allowed to slip. The details of the mechanism used for operat- 
ing the clutch are shown clearly in the figure. 



440 



HELE-SHAW CLUTCH 



[Chap. XVI 



Analysis of the Hele-Shaw clutch. — Since the surfaces of contact 
are frustums of cones, the action of the Hele-Shaw clutch is similar 




Fig. 241. 




to that of an ordinary cone clutch; hence the formulas derived in 
Art. 295 are applicable. In Fig. 242 are shown a pair of discs as 
used in the Hele-Shaw clutch. Applying the principles discussed 



Art. 308] IDEAL MULTI-CONE CLUTCH 441 

in Art. 295, we find that the magnitude of the moment of friction 
Mi for the frustum of the outer cone is 

M l = S^-' (451) 

4 sin a v ' 

and that on the inner cone is 

M , = f^ (452) 

4sma v ' 

The moment of friction for one friction surface is the sum of 
ilfi and M 2) and for s surfaces the total torsional moment that 
the clutch is capable of transmitting is given by the following 
expression : 

As now constructed, the number of discs used in the standard 
sizes of Hele-Shaw motor-car clutches is always odd, ranging 
from 15 to 33, and s in (453) is always one less than the total number 
of discs used. 

The horse power transmitted by a Hele-Shaw clutch may be 
calculated by means of the following formula: 

„ suPND ,._.. 

126,060 sin a 

in which D denotes the mean diameter of the discs as shown in 
Fig. 242, and N denotes the revolution per minute. 

308. Ideal Multi-cone Clutch. — A clutch of the conical-disc 
type having but one disc was recently placed on the market by 
The Akron Gear and Engineering Co., of Akron, 0. This clutch 
is shown in Fig. 243 in the form of a friction coupling, connecting 
shafts a and h. The driving shaft a has keyed to it a sleeve b 
to which the steel casting cone c is keyed. The internal surface 
of cone c comes in contact with the cone d, while the outer sur- 
face comes in contact with the conical bore of the casing g. The 
part of the clutch casing marked / is screwed onto the casing g, 
and is equipped with lugs on the inner surface. These lugs cause 
the cone d to rotate with /, and at the same time permit d to be 
moved in an axial direction by the operating mechanism. To 
provide adjustment for wear at the contact surfaces, the' cone d 
is screwed onto the ring e. This ring, held central by the casing 
/, is provided with a series of slots on its periphery, into which 
the set screws I may be inserted after the adjustment for wear 
has been made. 



442 



MOORE AND WHITE CLUTCH 



[Chap. XVI 



The axial pressure forcing the cones d, c, and g together is that 
due to a series of roller toggles that are operated by the sliding 
sleeve m. In disengaging the clutch, the rollers n are, moved 
towards the center of the shaft and come in contact with the 
raised part of the lugs o, which are cast integral with the ring e. 
As a result, the cone d is pulled out of engagement. Since the 
casing stands idle when the clutch is disengaged, it may be par- 
tially filled with oil, thus causing the driving cone c to run in oil 
and insuring good lubrication at the surfaces in contact. 

309. Moore and White Clutch. — In Fig. 244 is shown a friction 
coupling in which the disc c is fitted with hardwood blocks, the 



k d f 




Fig. 243. 



ends of which are brought into contact with the flanged hub d 
and ring e through the operation of the double toggle mechan- 
ism. Suitable lugs on the disc c engage corresponding recesses 
on the flange b, thus causing c to rotate with the latter, and at 
the same time permitting it to move in an axial direction. The 
surface of the wooden blocks in contact with d is flat, while the 
end in contact with the ring e is in the form of a double cone, as 
shown in the figure. The clutch is provided with a series of 
springs between the hub d and ring e, which prevent excessive 
wear of the friction surfaces when the clutch is disengaged. 
Another form of combined conical-disc clutch, known as a slip 



Art. 309] 



MOORE AND WHITE CLUTCH 



443 



gear, is shown in Fig. 152, and a description of it is given in Art. 
231. 

The relation between the design constant K 2 and the mean 
velocity of the friction surfaces for the type of clutch illustrated 
in Fig. 244 is shown graphically in Fig. 245. 




V 10 
o 

= 5 

> 






















V 












•». 































































































































































































































































































































































































































































































































500 600 

Mean Velocity 

Fig. 245. 



700 800 

ft per min. 



RIM CLUTCHES 



A large number of different forms of rim clutches are manu- 
factured, and apparently they vary only in the form of the rim 
or in the method of gripping the rim. A study of commercial 
rim clutches leads to the following classification: (a) block; 
(6) split-ring; (c) band; (d) roller. 



444 



EWART CLUTCH 
BLOCK CLUTCHES 



[Chap. XVI 



Block clutches are used chiefly on line shafts and counter- 
shafts, although there are several designs that have given good 
service on machine tools. Examples of the former type are shown 




Fig. 246. 

in Figs. 246, 247, and 248, while the latter type is represented in 
Fig. 249. 

310. Transmission Block Clutches. — (a) Ewart clutch — In Fig. 
246 are shown the constructive features of the well-known 




Fig. 247. 

Ewart clutch. The levers that move the friction . blocks are 
located inside of the clutch rim a, thus decreasing the air resist- 
ance at high speeds, and at the same time making it less dangerous 
to workmen than the type of clutch in which the operating levers 
and links are exposed. The Ewart clutch is fitted with either 



Art. 310] 



HUNTER CLUTCH 



445 



two, four, or six friction blocks, depending upon the power that 
is to be transmitted. 

(6) Medart clutch. — The type of clutch coupling shown in 
Fig. 247 differs from the Ewart clutch in that the friction blocks 
are of V shape. Furthermore, the operating levers are exposed, 
thus making this clutch more or less dangerous. For trans- 
mitting large powers, the Medart clutch is made as illustrated in 
Fig. 247, while for small powers, the clutch rim b is made flat. 

(c) Hunter clutch.' — The Hunter clutch coupling, shown in 
Fig. 248, has two cast-iron shoes c and d which are made to clamp 




the drum b when the screws g and h are rotated by the levers 
k and m. Each of the shoes is fitted with a driving pin, by means 
of which the shoes c and d are made to revolve with the flanged 
hub / and the shaft q. The holes in the flange /, through which the 
driving pins pass, are elongated in order that the shoes may move 
freely in a radial direction. The drum 6 is fastened to the shaft 
a by a feather key, thus permitting it to be drawn out of contact 
with the shoes c and d when the coupling is not transmitting 



446 



MACHINE-TOOL BLOCK CLUTCH [Chap. XVI 



power. The levers k and m are operated by the usual links and 
sliding sleeves as shown in Fig. 248. 

(d) Machine-tool block clutch. — In general, a block clutch used 
on machine tools consists of a shell running loose on the shaft, 
into which are fitted two brass or bronze shoes. These shoes 
are fastened loosely to a sleeve, which in turn is splined to 
the shaft. The shoes are pressed against the inner surface of 
the shell by means of an eccentric, screw, or wedge. Due to 
the compactness of such clutches, they are well adapted for use 
where the space is limited, as for example between the reversing 
bevel gears of a feed mechanism as shown in Fig. 249. In the 
design illustrated by Fig. 249, the enlarged bore of the bevel gears 
a forms the shell against which the shoes c and d are pressed by 




Fig. 249. 



the sliding sleeve /. This sleeve is integral with the double 
wedges e that are fitted to slide along the inclined surface of the 
shoes c and d, as shown in the figure. The friction shoes are 
fastened by filister head machine screws to the sleeves b and 
therefore rotate with them. Sleeve / is fastened to the shaft 
by means of a feather key. The shaft g may serve either as the 
driving or the driven member. 

311. Analysis of Block Clutches. — In order to arrive at an 
expression for the moment of the frictional resistance of a block 
clutch, some assumption regarding the distribution of the con- 
tact pressure, as well as the variation in the coefficient of fric- 
tion, must be made. As in the case of axial clutches, experi- 
mental data are lacking, and in what follows, we shall assume that 



Art. 311] 



BLOCK CLUTCH ANALYSIS 



447 



the normal wear at any point of the contact surface is proportional 
to the work of friction, and that the coefficient of friction remains 
constant. 

(a) Grooved rim. — For our discussion, we shall assume a block 
clutch in which the rim is grooved as shown in Fig. 250. The 
total moment that a clutch of this type will transmit is equal to 
the number of blocks in contact multiplied by the frictional 
moment of one block, the magnitude of which may be determined 
as follows: 

In Fig. 250 is shown a grooved clutch rim against which a 
single block is held by the force P; hence, the normal force acting 




Fig. 250. 



upon an elementary area of the surface in contact is prdddf, 
and the component of this pressure parallel to the line of action 
of the radial force P is given by the expression 



Hence 



dP = pr sin/3 cos# dd df 
P = 2 sinpffprcosd dedf 



(455) 



Since the normal wear n at any point is assumed to be propor- 
tional to the work of friction, we get 

n = kpr 

If the surfaces in contact remain conical, it follows that the wear 



448 BLOCK CLUTCH ANALYSIS [Chap. XVI 

h in a direction parallel to the line of action of P is constant ] 
hence, the normal wear is 

n = h sin /3 cos 6 
Combining the two values of n just given, we obtain the relation 

C cos d , ' 

p = ' (456) 

in which C = — t — Substituting (456) in (455), and integrating 

between the proper limits, the following expression for P is found : 

P = 2 C/sin/3 (0 + sin d cos 6) (457) 

The moment of the force of friction acting upon the elementary 
area is 

dM = npr*d6df = -^ rcosddddr. 
* cos /3 

The total moment per block is therefore: 

* ■- ^£-' (rj- rD («8) 



Combining (457) and (458) in order to eliminate the constant C, 
we obtain the following expression for the total moment per 
block : 

ijlPD r sinfl 



M 



vpjj r sine , . 

sin jS Id + sin 6 cos d\ y J 

Since (459) gives the magnitude of the frictional moment that each 
block will transmit, assuming that the radial force per block is 
P, the total moment that the clutch is capable of transmitting is 
obtained by multiplying (459) by the number of blocks. 

(b) Flat rim. — The majority of the block clutches in common 
use have a flat rim; hence making (3 = 90 degrees in (459), the 
frictional moment transmitted by each block becomes 

id + sin 6 cos 61 

The total moment is obtained in the same manner as outlined in 
the preceding paragraph. 

To facilitate using (459) and (460), the function in the brackets 
may be evaluated for different angles and the results thus ob- 
tained may be plotted. Fig. 251 gives values of . , — ^ t- 

for various values of 6. 



Art. 312] 



SPLIT-RING CLUTCHES 



449 





— 0.59n 


















































































- 7 


CD 
c 
'55 


<2> w-3o 

o 

u 0.57 
<D 

.E0.56 
in 

©0.55 








































7 








































-S — 








































sJL 






































/ 














































































/ 






































y 








q- 0.54 

o 

u) 0.53 

- 0.52 
c 

> 

0.51 




































































































































































































































































































































0.5- 











































20 



30 



40 



Values o-F 
Fig. 251. 



50 



Degrees 



60 



SPLIT-RING CLUTCHES 

312. Machine-tool Split-ring Clutches. — Split-ring clutches 
are used for all classes of service but their greatest field of applica- 
tion appears to be in connection with machine tools, or in places 
where the diameter of the clutch as well as the space taken up by 




Fig. 252. 



the clutch is limited. Such clutches are shown in Figs. 252 
to 254, inclusive. An inspection of these figures shows that in 



450 



SPLIT-RING CLUTCHES [Chap. XVI 




Fig. 253. 




Fig. 254. 



Art. 313] SPLIT-RING CLUTCH ANALYSIS 451 

general a split-ring clutch consists of an outer shell running loose 
on a shaft or sleeve; into this shell is fitted a split ring. The 
latter may be expanded by the action of a pair of levers as shown 
in Figs. 252 and 253, or by means of a wedge as shown in Fig. 
254. A sliding sleeve, operated by a suitable lever, forms a con- 
venient means of engaging the split ring with the outer shell. 
The outer shell may be in the form of a gear as shown in Figs. 
253 and 254, or it may form part of a pulley. 

The well-known Johnson clutch shown in Fig. 252 is used on 
countershafts and on machine tools. It has been adopted by 
several manufacturers of machine tools and other classes of 
machinery. The clutch shown in Fig. 253 is that used by the 
Greaves Klusman Tool Co. on their all-geared-head lathe. The 
split clutch represented in Fig. 254 is that used by the American 
Tool Works on the double back-gear of their high-duty lathe. 

313. Analysis of a Split -ring Clutch. — (a) Moment of friction. — 
For a split-ring clutch, it seems reasonable to assume that the 
pressure exerted by the ring upon the clutch shell is uniformly 
distributed over the area in contact; hence, the expression for the 
moment of the force of friction acting upon the elementary area is 

AM = ee@™, (46D 

in which D denotes the diameter of the split ring, / its face, and 
p the normal pressure per unit of area of the ring. 

The split ring has an angle of contact with the shell of some- 
what less than 360 degrees, but for all practical purposes we may 
assume it as equal to 360 degrees. Assuming the coefficient of 
friction n as constant, the total torsional moment transmitted 
by the clutch is obtained by integrating (461). Thus 

M=>^^ (462) 

(b) Horse power transmitted. — The horse power transmitted by 
the clutch at N revolutions per minute is 

_ ^*pfND* ( 

H ~ 396,000 (463) 

Since n and p are constant for any given case, their product may 
be denoted by a new constant, as i£ 4 . Hence 

B = KtfND ' (464) 

40,120 K ' 



452 



SPLIT-RING CLUTCH ANALYSIS 



[Chap. XVI 



(c) Force required to spread the split ring. — The inside diameter 
of the shell of a split-ring clutch is generally made ^4 to ^2 
inch larger than the diameter of the ring. Due to this fact, a 
certain part Pi of the force P exerted by the operating mechanism 
is used in spreading the ring. As soon as the ring comes into con- 
tact with the shell, a force P 2 is required which will press the ring 
against the shell, thereby causing the frictional moment necessary 
to transmit the desired power. The sum of Pi and P 2 must evi- 
dently equal the magnitude of the force P. 

1. Determination of Pi. — In the following analysis we shall 
assume that the thickness of the ring is small relative to its radius, 
and that the ring will readily conform to the bore of the shell. 



F> + P 2 



p \ + p z 




-f-r 



Fig. 255. 



According to Bach's "Elasticitat und Festigkeit," the moment 
of the force Pi about the section at A in Fig. 255 is given by the 
following expression : 

2P 1 r 1 = i?l[i-i], (465) 

in which ri and r 2 denote respectively the original and final radii 
of the ring. Therefore, the magnitude of Pi is 

P - EI 



riLr 2 rj 



(466) 



2. Determination of P 2 . — The pressure upon an elementary 

length of the ring is ^73 — , and the moment of this pressure about 

z 

the section at A in Fig. 255 is 



Art. 314] STUDY OF SPLIT-RING CLUTCHES 



453 



pfD* sin dd 
4 



Integrating between the proper limits, we find that the bending 
moment upon the ring at the section through A has the following 
magnitude : 

M A = ^ (467) 

Since this bending moment must equal that due to the force P 2 , 
it is evident that 




from which 



Mean Velocity -ft. per 
Fig. 256. 



P*D = 



P 2 = 



pfD 2 

2 

VfD 



(468) 



Combining (462) and (468), the magnitude of P 2 in terms of M, 
n, and D is as follows: 

M 
P* = ^ (469) 



314. Study of Split-ring Clutches. — From a study of a con- 
siderable number of split-ring clutches of different types, it was 
found that in nearly all cases the ring and shell are made of cast 
iron. In the majority of the designs, the ring is of the expanding 



454 



FARREL BAND CLUTCH 



[Chap. XVI 



type shown in Figs. 252 to 254, inclusive. The contracting-ring 
type is also used, but not to any great extent. An analysis, simi- 
lar to that made of the cone and disc clutches, was made of a 
number of split-ring transmission clutches. From the informa- 
tion furnished by two manufacturers, it was possible to determine 
the value of the design constant Ki for the various clutches. 
The graph AB of Fig. 256 represents the results obtained on 
five clutches of the contracting split-ring type made by one manu- 
facturer. The graph CD represents the results obtained on 
eleven clutches of the same type as the others, but made by 
another manufacturer. 




Fig. 257. 



BAND CLUTCHES 



Band clutches are usually installed when it is necessary to 
transmit heavy loads accompanied by shocks, as for example, in 
the drives of rolling mills and heavy mine hoists. In general, a 
band clutch consists of a flexible steel band, either plain or faced 
with wood or asbestos fabric, one end of which is fixed and the 
other is free to move in a circumferential direction. Due to the 
pull exerted by the operating mechanism on the free end of the 
band, the latter is made to grip the driving or driven member. 

315. Types of Band Clutches. — (a) Farrel clutch. — A band 
clutch in which the band is given several turns around the driving 
drum is shown in Fig. 257. In this design, the driving drum is 
keyed rigidly to the shaft a and both rotate in the direction indi- 
cated by the arrow. The unlined steel band e is given approxi- 



Art. 315] WELLMAN-SEAVERS-MORGAN CLUTCH 



455 



mately six and one-half turns around the drum g. One end of 
this band is fastened to the flanged hub b in the manner shown 
in Fig. 257(6), and the free end is operated by the special lever d 
through the medium of the conical ended shipper sleeve h. 

(6) Wellman-Seavers-M organ clutch. — Another form of single 
band clutch, installed on heavy mine hoists by the Wellman- 
Seavers-Morgan Co., is shown in Fig. 258. The band is lined 




Fig. 258. 



with wood and has an angle of contact on the clutch ring g of 
approximately 300 degrees. The flanged hub b, upon which the 
various parts of the clutch proper are mounted, is keyed to the 
driving shaft, while the hoisting drum, to which the clutch ring 
is bolted, runs loose on the shaft. 

(c) Litchfield clutch. — In Fig. 259 is shown a two-band clutch 
designed by the Litchfield Foundry and Machine Co. for use on 



456 



BAND CLUTCH ANALYSIS 



[Chap. XVI 



mine hoists. The bands are lined with wood and each band has 
an arc of contact with the drum g approximating 140 degrees. 

316. Analysis of a Band Clutch. — The principles underlying 
the design of a band clutch are similar to those employed in de- 




termining the power transmitted by a belt. In other words, the 
ratio of the tight to the loose tensions in the band or bands is 
given by the following expression: 






(470) 



Art. 317] 



MORTON ROLLER CLUTCH 



457 



in which T x and T 2 denote the tight and loose tensions, respec- 
tively, fi the coefficient of friction, and the angle of contact. 
The value of the coefficient of friction ju for a steel band on a cast- 
iron drum may be assumed as 0.05 when a lubricant is used, and 
0.12 when no lubricant is used. For a wood-faced band, /j, may 
be assumed as 0.3. 

ROLLER CLUTCH 

317. Horton Clutch.— The type of rim clutch shown in Fig. 260 
is known as the Horton roller clutch, and is used to some extent 




on punching presses. The cam a is keyed to the crankshaft, and 
upon its circumference are cut a number of recesses which form 
inclined planes. The rollers d, rolling up these inclined planes 
due to the action of the shell e, wedge themselves between a and 
the clutch ring b, thus causing the crankshaft to rotate with the 
flywheel. The ring b is keyed to the flywheel or the driving gear. 
The rollers are held in place and controlled by the shell e, which 
is connected with the crankshaft by means of a spring. The 



458 CLUTCH ENGAGING MECHANISMS [Chap. XVI 

latter is not shown in the 'figure. The lug / on the shell e engages 
a latch or buffer which is operated by the treadle o*n the machine. 
The method of operation of this roller clutch is as follows: 
At the instant the treadle releases the shell e, the spring rotates 
the latter around the shaft a short distance, carrying the rollers 
with it. This action causes the rollers to wedge between the 
cam a and the ring b, thus forming a rigid connection between 
the flywheel and the crankshaft. To disengage the clutch, the 
treadle is released and it in turn causes the latch or buffer to 
strike the lug /, thus forcing the cage and rollers back into the 
original position, and permitting the flywheel to rotate freely 
again. 

CLUTCH ENGAGING MECHANISMS 

318. Requirements of an Engaging Mechanism. — From the 
descriptions of the various types of clutches given in the 
preceding articles of this chapter, it is evident that clutches 
are engaged by a lever or shipper arm through the medium of an 
engaging mechanism which is capable of increasing the leverage 
rather rapidly toward the end of the lever displacement. In 
the analysis of the various types of clutches, the force required 
at the end of the operating lever was not discussed, since its 
magnitude depends directly upon the engaging mechanism. 

In Figs. 223, 248, 261, and 262 are shown four types of engag- 
ing mechanisms in which the following important requirements 
are fulfilled : 

(a) The arc of lever movement is not excessive. 

(6) The leverage increases rapidly toward the end of the lever 
displacement. 

(c) The engaging mechanism is self -locking, and therefore no 
pawl and ratchet are necessary to hold the clutch in engagement. 

(d) The force required at the end of the operating lever in 
order to engage the clutch is not excessive. The magnitude of 
this force may be assumed to vary from 15 to 20 pounds in the 
case of an overhead clutch installation. For large clutches 
these values may have to be increased somewhat. 

319. Analysis of Engaging Mechanisms. — In the majority 
of engaging mechanisms, graphical methods are generally found 
more convenient for determining the magnitude of the force 
required at the end of the operating lever. In Fig. 261(6) is 



Art. 317] ANALYSIS OF ENGAGING MECHANISMS 



459 



given the graphical analysis of the forces acting on the mechan- 
ism shown in Fig. 261(a). The vector A B represents the force 
Q exerted upon the spool b by the operating lever. Assuming 
that the clutch is provided with two toggle levers c, only one 
being shown in Fig. 261(a), the force exerted by b upon c 
is represented by the vector AC. The lever c is acted upon by 
three forces as shown in the figure. The magnitude of P is 
represented by the vector CD. The analysis of the forces acting 
upon the second toggle lever is given by the triangle BCE, in 
which the vector EC represents the magnitude of the force corre- 




sponding to P. The magnitude of the axial force produced is 
given by the vector ED. 

Analytical methods sometimes are found more convenient 
than graphical methods, as the following analysis will show. It 
is required to determine an expression for the force P in terms 
of Q in the case of the mechanism shown in Fig. 262(a). This 
mechanism is used on the split-ring clutch shown in Fig. 254. 
The sliding key g is acted upon by three forces as follows: Q, 
a on g, and / on g. In this case, Q denotes the force that the 
operating lever exerts upon the collar j shown in Fig. 254. 



460 



ANALYSIS OF ENGAGING MECHANISMS [Chap. XVI 



Taking components of Q and f on g in a, direction at right angles 
to a on g, we obtain the relation: 

Q cos <p = (f on g) sin (a + 2 <p) 

Q cos (p 



from which 



f ong 



(471) 



sin (a -\- 2 <p) 

At the upper end of the sliding wedge /, the ring e produces a 
force that will act vertically downward as shown in Fig. 262. 
This force has a magnitude given by the expression : 

F = 2P tan (/? + *>), (472) 




in which P denotes the magnitude of the force tending to spread 
apart the ring e. Taking components of F and / on g in a 
direction at right angles to d on f, we have 

F cos <p = (f on g) cos (a + 2 <p) 

from which we find that 

Q 



F = 



(473) 



tan (a + 2 <p) 

Combining (472) and (473), we obtain the following relation 
between P and Q : 



Q 



2 tan (a + 2 <p) tan (j8' + <p) 



(474) 



Art. 317] ANALYSIS OF ENGAGING MECHANISMS 461 

The graphical analysis for the mechanism illustrated by Fig. 
262(a) is shown in Fig. 262(6). The vector AB represents the 
magnitude of the force Q. The force F is represented by CD, 
and P by the vector FE. 

For the analysis of a screw operated mechanism, consult 
Art. 298. 

References 

Die Maschinen Elemente, by C. Bach. 

Machine Design, Construction, and Drawing, by H. J. Spooner. 

The Gasoline Automobile, by P. M. Heldt. 

Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 

Mechanical Engineers Handbook, by L. S. Marks, Ed. in Chief. 

Clutches with Special Reference to Automobile Clutches, Trans. A.S.M.E., 
vol. 30, p. 39. 

Friction Clutches and Their Use, Power, Apr. 11 and May 2, 1911. 

Friction Clutch and Operating Gear for Cruising Engines and Turbines, 
Jour. A. S. of Mar. Engr., vol. 26, p. 206. 

Couplings for Cruising Turbines, London Eng'g., July 4, 1913. 

Coil Friction Clutches, Amer. Mach., Apr. 1, 1909. 

Friction Clutches, Proc. Inst, of Mech. Engr., 1903. 



CHAPTER XVII 
BRAKES 

The function of a brake is to absorb energy by the creation of 
frictional resistance, and thereby reduce the speed of a machine 
or bring the machine to a state of rest. The absorbed energy 
must equal that given up by the live load and all moving parts 
that are being retarded. Friction in bearings and between other 
moving parts always helps a brake. 

320. General Equations. — The energy absorbed by a brake is 
made up of the following factors: (1) The work given up by the 
live load; (2) the energy given up by the rotating parts. To 
determine an expression for the tangential force required on the 
brake sheave so as to bring a load to rest, we shall assume the 
case of a geared hoisting drum lowering a load. 

Let D = diameter of the drum. 
W = the load on the drum. 
T = tangential force on the brake sheave. 
d = diameter of the brake sheave. 
n = ratio of the gearing between the drum and the 

brake sheave. 
t = number of seconds the brake is applied. 
v = linear velocity of the load in feet per second. 
7] = efficiency of the mechanism. 

To bring the live load to a stop in i seconds requires an expendi- 

Wv Yv "1 
ture of -~- - + t\ foot-pounds of work at the drum. In addition 

to absorbing the work due to the live load, the brake in bringing 
the machine to a stop must also absorb the kinetic energy of all 
of the rotating parts. The energy due to the rotating parts is 
§ Ico 2 , in which I is the moment of inertia of the rotating parts 
referred to the axis of the brake, and w is the angular velocity of 
these parts in radians per second. It is usually possible to obtain 
the value of \ Jw 2 for a rotating mass having a complicated form. 
The body may be divided into elements in such a way that the 

462 



E = v — + t: + ~o~ ( 475 ) 



Art. 320] GENERAL EQUATIONS 463 

kinetic energy of each element is easily calculated; then by 
summation the total kinetic energy is obtained. 

Taking into account the internal friction of the machine, the 
total energy to be absorbed by the brake in t seconds is given by 
the following expression: 

Wvt . Wf . 1^ 

ndtvT 
The work done by the tangential force T in t seconds is ~ j^ 

foot-pounds. Since the energy given up by the load and rotating 
parts must equal that absorbed by the brake, we have 

The minimum value of the force T on the brake sheave occurs 
when the load has been brought to a state of rest and its magni- 
tude is evidently 

'-^ 

321. Classification. — Brakes are made in a variety of forms and 
no definite classification can be given. The order in which they 
are discussed in this chapter is as follows: 

(a) Block brakes. 

(6) Band brakes. 

(c) Axial brakes. 

(d) Mechanical load brakes. 



BLOCK BRAKES 

322. Single- and Double-block Brakes. — The common form 
of block brake has a single block pressing against the sheave, thus 
causing an excessive pressure upon the shaft bearings. Such a 
brake is shown in Fig. 263. The pressure upon the shaft bear- 
ings caused by the block in this form of brake may be practically 
eliminated by the use of two brake blocks located diametrically 
opposite to each other. An arrangement of this kind is used on 
cranes, elevators, and mine hoists. In Figs. 264 to 267, inclusive, 
are shown various designs of double-block brakes all of which are 
drawn to scale. The brakes illustrated by Figs. 266 and 267 
are used on mine hoists, and are commonly called post brakes. 
The first of these post brakes was designed and built by an English 



464 



BLOCK BRAKES 



[Chap. XVII 



manufacturer for a large mine hoist; while the second, designed 
and built by the Nordberg Engineering Co., is used on a large 




hoist installed at the Tamarack mine at Calumet, Mich. As 
shown in Fig. 267, the posts of the Nordberg brake are held in 




position by the swinging links m, n, and o. The blocks or shoes 
are made of steel casting, instead of wood as in the English design 
shown in Fig. 266. 



465 




Fig. 265. 




Fig. 266. 



466 



BLOCK BRAKE ANALYSIS 



[Chap. XVII 



323. Analysis of Block Brakes. — In determining the magnitude 
of the forces coming upon the various members of a block brake, 
either algebraic or graphical methods may be used. Frequently 
the latter save considerable time. 

Let F = the force acting at the fulcrum of the operating 

lever. 
K = the force applied at the end of the operating lever. 
P = the radial force exerted by the sheave upon the 

block. 
T = the tangential resistance upon the block. 
ix = the coefficient of friction. 




Since the action of the block upon the brake sheave is similar 
to the action between the shoes and sheave of a block clutch, the 
various formulas derived in Art. 311 may be applied directly. 
Two cases will be considered, the first involving the action of a 
grooved sheave and the second the action of a flat sheave. 



Art. 323] BLOCK BRAKE ANALYSIS ' 467 

(a) Grooved sheave, — A block brake having a grooved sheave 
is shown in Fig. 263. The moment of the frictional resistance 
due to the radial force P is given by the following expression: 

M = >** T Sin ^ 1 (478) 

sin /3 L 6 4- sin cos el l ' 

The product of the factors u and ^—i — = — 7, 7, may be treated 

e + sm e cos e J 

as a new factor denoted by the symbol yJ. This factor might 
be termed the apparent coefficient of friction between the brake 
block and sheave. 

To determine an expression for the tangential resistance T, 
divide the moment M by the radius of the sheave, thus 

T = —^ (479) 

sm $ K ' 

To determine an expression for the force K applied at the end 
of the operating lever, treat the latter as the free body and take 
moments about the fulcrum. Thus for the rotation indicated 
in Fig. 263, we have 

a v 

To calculate the size of the pin at the fulcrum, we must de- 
termine the magnitude of the force F coming upon the pin. In 
general, the graphic analysis affords the most direct means of 
determining the magnitude of F. Treating the brake lever as 
the free body, the magnitudes of the various forces are readily 
obtained by drawing the force diagram ABDCA, in which the 
vectors BC, CA, AD, and DB represent the forces F, K, P, and 
T, respectively. Having determined the magnitude of the force 
F, the pin at the fulcrum must be proportioned so that it is 
capable of resisting the bending moment and bearing pressure 
coming upon it. 

In the above analysis, the frictional resistance on the brake 
shaft was not considered. The error in any case is small and 
the method given is the one commonly used. However, when 
designing the bearings on the brake shaft, the pressure due to 
the force P should not be neglected. 

(b) Flat sheave. — In the majority of installations, the brake 
sheave is made with a straight or flat face; hence in the preceding 
expressions for M and T the angle becomes 90 degrees. 
Substituting this value in (479), we get 

T = 2 n'P (481) 



468 BLOCK BRAKE ANALYSIS [Chap. XVII 

The magnitudes of K and F for this case are obtained in a manner 
similar to that given above. 

To facilitate the calculations for the value of the apparent 
coefficient of friction fj,', the graph given in Fig. 251 will be found 
useful. 

324. Graphical Analysis of a Double-block Brake. — It is re- 
quired to determine graphically the magnitude of the force re- 
quired to apply the double-block crane brake shown in Fig. 
264, assuming that the total moment of the frictional resistance 
on the brake sheave is known. The following method of 
procedure is suggested : 

(a) Determine the relation between T and P for each of the 
blocks by means of (481); thus 

^=g = 2/ (482) 

Having calculated the value of p', for the brake under discussion, 
the actual lines of action of the resultant of T and P on each block 
may be laid off as shown in Fig. 264(6). In addition to the re- 
sultant of T and P, each brake block is acted upon by another 
force which is equal to the resultant but acts in the opposite 
direction. From this it is evident that each brake block exerts 
a pressure upon its lever equal to the resultant. 

(6) Each brake lever is acted upon by three forces, as follows : 
(1) A pressure due to the spring shown in the figure; (2) the force 
due to the brake block; (3) the reaction at the fulcrum of the 
lever. Of the three forces just mentioned, the lines of action of 
the first two are known, also the magnitude of the second. Since 
the point of application of the third force is known, the remaining 
unknown properties of the forces acting upon the levers may 
readily be determined. Applying the "triangle of forces" to 
the left-hand lever, the magnitudes of the various forces acting 
thereon are represented by the sides of the triangle ABC of Fig. 
264(6). The vector CA represents the magnitude of the spring 
pressure K x upon this lever, as well as upon the right-hand lever. 
The vector CB represents the magnitude of the pressure exerted 
upon the fulcrum by the left-hand lever. 

Applying the conditions of equilibrium to the forces acting upon 
the right-hand lever, we find that a couple is necessary to produce 
equilibrium. Since Ki = K 2 and Ri = R 2 , it is evident that the 
resultant pressure upon the fulcrum, due to the right-hand lever, 
is equal to that produced by the left-hand lever, but it acts in the 
opposite direction as shown in Fig. 264(6). 



Art. 325] 



SIMPLE BAND BRAKE 



469 



(c) Having determined the pressures upon the fulcrum, the 
dimensions of the pin may be calculated. 

(d) The operating lever a, which is actuated by a solenoid, is 
used to disengage the brake. Knowing the spring pressure, 
the magnitude of the force K 3 is readily obtained by applying 
the " triangle of forces." 

BAND BRAKES 

In a band brake an iron or steel band, lined with wood, leather, 
or asbestos-fabric encircles the brake sheave and is so arranged 
that it may be tightened or released. There are three types of 




band brakes, as follows: (a) simple; (&) band brake for rotation 
in both directions; (c) differential. 

325. Simple Band Brakes. — Two different designs of simple 
band brakes are shown in Figs. 268 and 269. In general, brakes 
of this type should be designed so that the heaviest pull will 
always come upon the anchorage. A band brake arranged in 
this manner requires a comparatively small pull at the free end 
of the band to apply the brake. In the brake shown in Fig. 269, 
the band makes more than a complete turn about the sheave and 
for that reason the effort required to apply such a brake is small. 

Force analysis. — The following method of procedure for deter- 



470 



SIMPLE BAND BRAKE 



[Chap. XVII 



mining the force K required at the end of the operating lever 
is common to all of the simple band brakes. 

In Fig. 268 assume that the brake sheave rotates in the direc- 
tion indicated by the arrow (1). From Art. 122, the ratio of the 
tight to the loose tension is 

(483) 



k - -> 



in which y. and 6 denote the coefficient of friction and angle of 
contact, respectively. The net tension on the brake sheave is 



T = T 1 - T 2 = W-l) 



(484) 




Treating the operating lever as the free body, and taking 
moments about the fulcrum, we get 

X_™ = ** (485) 

a a(e» — 1) 

To determine the net cross-sectional area A of the band, divide 
the maximum tension 7\ by the permissible stress S of the mate- 
rial used : or 

A - l s b£r] (486) 

In calculating the dimensions of the band, the thickness should 



Art 326] 



REVERSIBLE BAND BRAKE 



471 



not be made so great that a considerable part of the force on the 
operating lever is used in overcoming the resistance to bending 
of the band. 

The pin at the anchorage or fulcrum must be made of ample 
size to resist the bending moment and pressures coming upon it. 

The analysis for the rotation indicated by the arrow (2) is 
similar to that just given, and is left to the student. 

326. Band Brakes for Rotation in Both Directions. — As men- 
tioned in the preceding article, the heaviest loaded band should 




Fig. 270. 

always be connected to the anchorage. This condition can readily 
be fulfilled provided the load acts continuously in one direc- 
tion. When the load reverses in direction, as in mine hoists, 
cranes, and elevators, the condition cannot be fulfilled. The 
design of a band brake, used on mine hoists and on the armature 
shafts of motors direct-connected to hoisting drums, is shown in 
Fig. 270. The two ends of the band are connected to the oper- 
ating lever at points which are equidistant from the fulcrum, as 
shown by the dimensions b. The force analysis of this type of 
brake is similar to that given in Art. 325. 

327. Differential Band Brakes. — The general arrangement of 
a differential band brake is shown in Fig. 271. As in the brake 



472 



DIFFERENTIAL BAND BRAKE 



[Chap. XVII 



illustrated by Fig. 270, both ends of the band are connected to 
the operating lever, but at different distances from the fulcrum. 
{a) Force analysis. — Assuming counterclockwise-rotation, the 
magnitude of the force K is given by the following expression : 

If in (487) the dimension c is made less than the product be 1 * 9 , the 
force K has a negative value and the brake is applied automatic- 
ally. This special condition is used to a considerable extent in 
connection with automatic crane brakes. In such installations, 

the automatic band brake is 
used to take the place of the 
ordinary ratchet and pawl 
mechanism. In Figs. 277 and 
278. are shown automatic band 
brakes fulfilling the function of 
a ratchet and pawl by permit- 
ting rotation of the brake sheave 
in only one direction. 

AXIAL BRAKES 

In the so-called axial brakes a 
frictional surface of revolution 
is forced against a correspond- 
ing surface, the pressure being 
applied in a direction parallel 
to the axis of rotation. Accord- 
ing to the form of the surfaces 
of revolution in contact, axial brakes may be divided into the 
following types: (a) conical brakes; (6) disc brakes. 

328. Conical Brakes. — One of the simplest forms of axial 
brake is the conical type, the constructive features of which are 
shown in Fig. 272. The magnitude of the force K at the end of 
the operating lever may be determined as follows. 

We shall assume that the outer cone forms a part of, or is 
attached to, the frame work of the machine, while the inner cone 
is splined to the rotating shaft. The inner cone is acted upon by 
the axial force Q and the pressure exerted by the outer cone upon 
the conical surface. The action of the conical brake is similar 




Fig. 271. 



Art. 328] 



CONICAL BRAKES 



473 



to the action of the cone clutch discussed in Art. 295. According 
to (426), the moment of friction that the brake is capable of ab- 
sorbing is given by the expression 

nQD 



M = 



(488) 



2 sin a 
from which the tangential resistance upon the cone becomes 

T = S~ (489) 



sin a 



Treating the operating lever as the free body and taking mo- 
ments about the fulcrum, we get 




K = 



Qb Tb sin a 



p.n 



(490) 



In some conical brakes the two cones are made of cast iron, 
while in others, such as are used on the armature shafts of crane 
motors, the outer cone is of cast iron and the inner cone is faced 
with wood. The angle a varies from 10 to 18 degrees, and the 
coefficient of friction may be assumed as 0.12 to 0.25. The 
former coefficient is to be used when both friction surfaces are 
made of cast iron, and the latter when one of the surfaces is of 
wood and the other of cast iron. 

329. Disc Brakes. — The disc brake is simply a special form of 
the conical brake having the cones opened out into plane discs. 
In practically all installations of disc brakes, there are more than 
two surfaces in contact. A disc brake having several contact 
surfaces is commonly called a Weston washer brake. Fig. 273 



474 



DISC BRAKES 



[Chap. XVII 



shows one form of multiple-disc brake. The pinion a, having 
a faced surface at b, is bushed and runs loose on the shaft c. The 
flange d, keyed to the shaft, has a faced surface similar to that 
at b. The shaft c carries a ratchet wheel which engages with a 
pawl and permits rotation in but one direction. Neither the 
ratchet wheel nor the pawl are shown in the figure. Between 
the faced surfaces of the pinion and the flange d, a series of fric- 
tion discs is arranged in such a way that alternate discs rotate 
with a and d. The discs shown in Fig. 273 are of fiber and steel; 
the former are keyed to the pinion by the feather keys e and 
the latter are fitted to the squared hub of the flange d. Fre- 
quently alternate discs of brass or bronze and steel are used. 




Fig. 273. 



The disc brake shown in Fig. 273 is used in connection with 
hoisting machinery when it is required to lower a load rapidly 
and have it under the control of the operator. For this reason 
it is frequently called a "dispatch brake." The shaft being held 
from running backward by the ratchet and pawl, the operator 
may lower the load by merely unscrewing the handwheel g. 
This action decreases the friction between the discs and at the 
same time releases the pinion. By screwing up the handwheel 
so as to increase the frictional resistance between the discs, the 
speed of the load may easily be controlled. In hoisting the load 



Art. 329] DISC BRAKES 475 

the handwheel, the flange d, and the pinion are locked together; 
in other words, the brake is converted into a clutch. 

Force analysis. — To determine the axial thrust Q and the 
force F on the rim of the handwheel that are necessary to apply 
the brake, the following analysis may be used. 

From (442) the moment of the frictional resistance of the 
discs is 

M = ^ (491) 

in which s denotes the number of friction surfaces and D the 
mean diameter of these surfaces. The axial thrust required to 
set the brake is 

To produce the axial thrust by means of the screw and hand- 
wheel, it is necessary to apply a force F on the rim of the latter. 
The moment of the force F must exceed the frictional moment 
of the screw plus the moment of friction between the flange d 
and the handwheel. Hence 

^^tan(a + /) +M', (493) 

in which D' denotes the diameter of the handwheel; a the angle 
of thread in the screw; <p f the angle of friction of the thread; and 
M' the frictional moment between d and g. 

The moment M a due to the load on the pinion, must overcome 
the pivot friction between the pinion and the thrust washer h, 
the journal friction between the pinion and the shaft c, and the 
moment of friction of the discs. Denoting the moment of pivot 
friction by Mi and that of journal friction by M 2 , the magnitude 
of M a is given by the following expression: 

M a = M + M x + M 2 (494) 

Substituting the value of M from (494) in (492) , we may calculate 
the magnitude of the thrust Q. In order to determine the force 
F substitute the value of Q in (493). 

For values of the coefficient of friction to be used in designing 
disc brakes those given in Art. 334(f) are recommended. 



476 LUDER'S BRAKE [Chap. XVII 

MECHANICAL LOAD BRAKES 

Mechanical load brakes are used chiefly in connection with 
chain hoists, winches, and all types of crane hoists. In general, 
the functions of a mechanical load brake are as follows: 

(a) The brake must permit the load to be raised freely by the 
motor. 

(o) It must be applied automatically by the action of the load 
as soon as the lifting torque of the motor ceases to act in the 
hoisting direction. 

(c) It must permit the lowering of the load when the motor 
is reversed. Reversing the motor releases the frictional resistance 
and allows the load to descend by gravity. 

Mechanical load brakes, also called automatic brakes, are made 
in a variety of forms, but the greater number used on modern 
cranes are of the disc type. 

330. Worm-gear Hoist Brakes. — In Fig. 274 are shown the 
constructive features of two forms of load brakes used on the 
worm shaft of German types of worm-geared chain hoists. These 
brakes are necessary to prevent the running down of the load, 
as the steep thread angle used on worms brings the efficiency of 
the hoist above 60 per cent. ; hence the worm and its gear are no 
longer self-locking. Brakes similar to the one shown in Fig. 
274(a) are also used on some American worm-geared types of 
drum hoists. 

(a) Luder's brake. — In Fig. 274(a) is shown a sectional view 
through Luder's automatic disc brake. The flanged hub b and 
the cap c are keyed to the end of the worm shaft. Between b 
and c, and rotating upon the hub of the latter, is a hollow bronze 
ratchet wheel e which engages with a pawl. The latter is not 
shown in the figure. The ratchet wheel is made hollow so as to 
form a convenient reservoir for the lubricant. Between 6 and 
e a leather or fiber friction disc is used. 

In raising the load the friction between the contact surfaces 
of e, due to the thrust of the load on the worm, is greater than 
that on the hard steel pivot /, and as a result the parts b, e, and 
c rotate with the shaft a. In lowering the load a pawl engages 
the ratchet wheel and holds it stationary, while the collar 6 
and the cap c rotate with the shaft, thus introducing extra fric- 
tion on both sides of e. The moment of the frictional resistance 
on e is made of such a magnitude as to prevent the overhauling 



Art. 330] 



BECKER'S BRAKE 



477 



of the load and still not make the pull on the hand chain too 
excessive. 

(6) Becker's brake. — The constructive features of Becker's 
automatic conical brake are shown in Fig. 274(6). In raising 
the load the friction between the cones b and c due to the thrust 
of the worm shaft a is greater than that between the screw / 
and the cap c; hence the latter rotates with the shaft, and the 
moment of friction is reduced to a minimum. In lowering the 
load, a pawl, not shown in the figure, engages the ratchet teeth 
e and prevents the cap c from turning; thus the moment of fric- 
tion caused by the thrust on the shaft a is that due to the two 
cones b and c. The moment of friction of these cones must be 
made sufficient to prevent the running-down of the load, and 
very little effort is required to lower the load by means of the 
pendant hand chain. 




Fig. 274. 



(c) Force analysis of Becker's brake. — It is required to deter- 
mine an expression for the mean diameter D of the cone in order 
that the hoist shall be self-locking, and further to determine an 
expression for the moment (P)R that is required on the hand 
sheave in order to lower the load. 

Let Q = the tangential load on the worm gear. 
R = the radius of the hand sheave. 
d = the mean diameter of the worm. 
a = the angle of the mean helix of the worm. 
6 = the half cone angle. 
<p' = the apparent angle of friction for the worm. 

To prevent the running-down of the load, the moment of the 
frictional resistances on the worm shaft must exceed the moment 
on the shaft a due to the load Q; hence . 



2sin0 + il ^T tan( *-« /) ' 



(495) 



478 NILES BRAKE TChap. XVII 

in which M x denotes the moment of friction on the shaft bearings. 
The magnitude of M x may be determined provided the diameter 
of the shaft and the distance between the bearings are known. 
However, this moment is generally small and for practical pur- 
poses may be neglected. Solving for D in (495), we have 

^ v. d tan (a — cp') sin 9 2 Mi sin d , Ar%n . 

D r~ mC - (496) 

The relation for D given by (496) must be fulfilled if the hoist is. 
to be self-locking. 

The moment on the hand chain sheave required to lower the 
load, assuming the hoist as self-locking, is 

{P)R = 5Wo ~ ? tan (a -^ + Mf > (497) 

in which M f denotes the frictional moment of the shaft bearings. 
The magnitude of M f may be determined approximately if the 
diameters of the shaft bearings are known. If ball bearings are 
used on the worm shaft, the loss due to the journal friction will 
probably not exceed 3 per cent, of the total work expended. 
Upon the latter assumption (497) reduces to 

331. Crane Disc Brakes. — (a) Niles brake. — In Fig. 275 is 
shown the design of a mechanical load brake used on cranes 
manufactured by the Niles-Bement-Pond Co. The spur gear a 
meshes directly with the motor pinion and is keyed to the sleeve 
b, which rotates freely upon the shaft c. The one end of this 
sleeve 6 is in the form of a two-jaw helical clutch mating with 
corresponding helical jaws on the collar h. The latter is keyed 
to the shaft, and to prevent it from sliding along the shaft c an 
adjustable thrust collar I is provided. The other end of the sleeve 
b is faced and bears against the phosphor-bronze disc /. A simi- 
lar disc g is located between the ratchet wheel d and the flange e. 
The latter is keyed to the brake shaft. The ratchet wheel d is 
bronze bushed and is free to rotate during the period of hoisting 
the load, but pawls, not shown in the figure, prevent rotation of 
d during the period of lowering the load. The pinion p meshes 
with the drum gear. 

To hoist the load, the motor rotates the gear a and the sleeve 6 
in the direction indicated by the arrow, while the shaft c, due to 



Art. 331] PAWLINGS AND HARNISCHFEGER BRAKE 



479 



the action of the load, tends to turn in the opposite direction. 
The relative motion between the helical jaws formed on b and h 
forces the sleeve b toward the flange e, thus locking the whole 
mechanism to the driving sleeve b. To lower the load, the motor 
rotates the sleeve b in a direction opposite to that indicated by 
the arrow, thereby reducing the pressure between the disc and the 
ratchet wheel. Releasing the thrust on the discs / and g permits 
the load to descend by gravity. As soon as the speed of the shaft 
c and the collar h exceeds that of the sleeve b, the relative motion 
between the helical jaws will cause an increase in the axial thrust 
between the discs and the ratchet wheel, which in turn locks the 
brake, since the wheel d is held by pawls. 



yy?:7:^Ti 




YZZ//ZZZZZ2 



p%M^ 



Fig. 275. 



(b) Pawlings and Harnischfeger brake. — The constructive fea- 
tures of a load brake used by the Pawlings and Harnischfeger 
Co. is shown in Fig. 276. It differs from the Niles brake in that 
the friction discs / and g are made of fiber instead of bronze, and 
instead of using a helical jaw clutch to produce the thrust upon 
the friction surfaces, the shaft c is threaded as shown. The 
driving spur gear a, which meshes with the motor pinion, has the 
bore of its hub b threaded so as to form a good running fit with the 
thread upon the shaft c. In general, the description and method 
of operation given for the Niles brake in the preceding para- 
graphs also apply to the brake shown in Fig. 276. 



480 



CASE BRAKE 



[Chap. XVII 



(c) Case brake. — Several crane manufacturers are using load 
brakes equipped with more than two friction discs. In Fig. 277 
is shown the design used by the Case Crane Co. The spur gear 




Fig. 276. 



a meshes directly with the motor pinion and is keyed to a flanged 
sleeve b, the bore of which is threaded so as to form a good work- 




ing fit with the thread on the shaft. The flange of the sleeve b 
bears against the first of the bronze friction discs /. The flanged 
hub e is keyed to the shaft c and bears against the last of the 



Art. 331] 



CASE BRAKE 



481 



bronze discs. The cast-iron friction discs g are keyed loosely to 
the hub e, while the discs /are keyed loosely to the shell d. The 
latter rotates freely during the hoisting period, but during the 
lowering of the load a properly proportioned differential band 
brake k prevents rotation. 

To hoist the load, the motor rotates the gear a and the sleeve 
b in the direction indicated by the arrow, while the shaft c, due 
to the action of the load, tends to turn in the opposite direction. 
Due to this relative motion, the threaded sleeve b will tend to 
screw up on the shaft, thus clamping the flanges b and e to the 
shell d. In this manner the whole mechanism is locked to the 
driving gear a, thus transmitting the required power to the pinion 
p. To lower the load, the motor rotates the gear a in a direction 




opposite to that indicated by the arrow, thus tending to reduce 
the axial thrust on the discs / and g and permitting the load to 
descend by gravity. Should the speed of the shaft c, due to the 
action of the load, exceed that of the gear a, the resultant relative 
motion will cause the sleeve b to screw up on the shaft and lock 
the brake, since the reverse rotation of the shell d is prevented by 
the differential band brake k. 

The closed shell d is made oil tight, thus assuring lubrication 
of the friction surfaces, since the discs run in an oil bath. In 
order to distribute the oil to the engaging surfaces all of the discs 
as well as the flanges b and e are provided with holes and grooves. 
Effective means of lubricating the screw threads are also provided, 
as shown in the figure. 

(d) Shaw brake. — In Fig. 278 is shown the design of an auto- 
matic multiple-disc brake used by the Shaw Electric Crane Co. 



482 



SHAW BRAKE 



[Chap. XVII 



The spur gear. a meshes directly with the motor pinion and is 
keyed to the sleeve b, which rotates freely on the shaft c. One 
end of the sleeve b is in the form of a two-jaw helical coupling 
mating with corresponding helical jaws formed on the flanged 
hub e. Cast integral with the sleeve b is the flange h, the inner 
face of which bears against the first of the cast-iron friction discs 
/. The flanged hub e is keyed to the shaft c and bears against 
the first of the cast-iron discs g. The discs / and g have lugs 
upon their outer circumferences which fit into recesses in the 
shell d and hence must rotate with d. The discs m and n have 
lugs upon their inner circumferences which fit into recesses on the 
sleeve b and hub e, respectively. The shell d rotates freely dur- 
ing the hoisting period, while during the lowering of the load a 
differential band brake located on the part k prevents rotation. 




Fig. 279. 



The operation of the Shaw brake is similar to that given in detail 
for the Case brake. An inspection of Fig. 278 shows that the 
engaging frictional surfaces may be run in an oil bath, and hence 
no trouble should be experienced as far as lubrication is concerned. 

332. Crane Coil Brakes. — A form of automatic coil brake using 
a continuous shaft is shown in Fig. 279. This design has been 
used successfully on cranes made by Niles-Bement-Pond Co. 
It consists of a shell a carrying at its closed end a ratchet wheel 
b engaging the pawls k. One end of the bronze coil d is fixed by 
means of lugs to the driving head c, and the other end is fixed to 
the driven head e. The driving head c, as well as the driving gear 
/, is keyed to the sleeve g which rotates freely on the shaft h. The 
driven head e is keyed to the shaft h and is provided with a lug 
that may engage with a similar lug on the sleeve g. These lugs 
perform the function of establishing a positive drive between the 



Art. 332] 



COIL BRAKE 



483 



sleeve g and the shaft h in case the bronze coil d wears down too 
far or in case the coil breaks. 

In hoisting the load, the gear / meshing directly with the motor 
pinion rotates the head c as shown by the arrow (1), while the 
driven head e and shaft h under the action of the load tend to 
turn in the opposite direction, thus expanding the coil d against 
the inner surface of the shell a. As a result of expanding the 
coil d, the whole mechanism is locked to the driving head c. The 
motor, in lowering the load, pulls one end of the coil until the 
contact surface between a and d is reduced sufficiently to enable 
the load to overcome the frictional resistance, thus permitting 
the load to descend by gravity. It should be remembered that 
the shell a is prevented from rotating in the reverse direction by 
the ratchet and pawls. The speed of lowering cannot exceed 
that due to the motor or the coil will expand and apply the brake. 




Fig. 280. 



333. Cam Brake. — The automatic cam brake shown in Fig. 280 
was designed to replace a troublesome coil brake of the two-shaft 
type. The shell a runs free on both of the shafts g and h. Upon 
the closed end of the shell is formed the ratchet wheel b, and a 
suitable pawl prevents rotation of the shell a when the load is 
lowered. The bronze coil originally used was replaced by two 
brass wings d, each of which has an arc of contact with the shell 
of about 165 degrees. A spider e, to which the wings are pivoted, 
is keyed to the driving shaft g, and the cam c which engages with 
these wings is keyed to the driven shaft h. 

In hoisting, the shaft g rotates as shown by the arrow (1), 
while the pinion shaft h under the action of the load tends to 
rotate in the opposite direction, thus causing the cam c to force 
the wings d outward against the shell and thereby locking the 
complete mechanism to the driving shaft g. In lowering, the 



484 ANALYSIS OF THE SHAW BRAKE [Chap. XVII 

rotation of the driving shaft is reversed, thus tending to release 
the wings d from between the casing a and the cam c, and permit- 
ting the load to descend by the action of gravity. As soon as 
the load tends to run down too fast, the cam forces the wings 
outward and automatically applies the brake. 

334. Force Analysis of an Automatic Brake. — In determining 
the relations existing between the external forces and the internal 
resistances acting on an automatic brake, the following analysis 
applied to the multiple-disc brake shown in Fig. 278 may serve 
as a guide. 

Let D = the mean diameter of the friction discs. 

I = the moment of inertia of the rotating parts located 
between the load and the brake, referred to the shaft 
of the latter. 
Q = the axial thrust on the helical jaws during hoisting 
period. 
(Q) = the axial thrust on the helical jaws during lowering 
period. 
R = the pitch radius of the hoisting drum. 
W = the load on the hoisting drum. 
a = the acceleration of the load while hoisting. 
(a) = the acceleration of the load while lowering. 
d = the mean diameter of the helical jaws. 
n = the gear ratio between the brake and the drum. 
a = the angle of the helical surface on the jaws. 
ip = the angle of friction for the helical surfaces. 
li — the coefficient of friction for the discs. 
r\ = the efficiency of the transmission between the brake 
and the load. 

(a) Axial thrust on helical jaws for hoisting. — During the hoist- 
ing period the action of the brake is similar to that of a clutch; 
hence the moment M required on the gear a in order to raise the 
load is 

M =\W + ^]K+^ (499) 

L g Jru) R 

Equating this moment to that of the internal resistance of the 
discs / and m, we get 

6MD '• f tan.(«.+ ,0 (590) 



Art. 334] ANALYSIS OF THE SHAW BRAKE 485 

Combining (499) and (500), we obtain the following expression 
for the axial thrust : 

2 R _ , Wa , 2 Ian 

W -\ 1 -p- 

Q = 5v.D + d tan (« + <p') (501) 

(b) Condition for self-locking. — When the power is shut off, 
the load W tends to run the brake and motor in the reverse 
direction. To prevent reversed rotation it is necessary that the 
moments of the frictional resistances of all of the discs and the 
several journals shall exceed by a small amount the moment 
due to the load. The moment of the load for the running down 

WR-q 

condition is , and this must be somewhat less than the 

n 

moment of friction of the discs / and g and the shell d, or 

^^ < 5 nQ'D, (502) 

n — 

in which Q' denotes the axial thrust on the helical jaws. The 
magnitude of Q' may be determined from (501) by making the 
acceleration a equal to zero; hence 

n = 2WR , , 

V nr?(5 »D + d tan (a + /)) K } 

To determine the relation that must exist between the dimen- 
sions of helical jaws and the friction discs so as to satisfy the 
condition of self-locking, combine (502) and (503); whence 

d tan (a + <p') < ^ (2 - 7j 2 ) (504) 

The relation expressed by (504) must be satisfied if the brake is 
to hold the load from running down. 

(c) Axial thrust on helical jaws for lowering. — If the power is 
shut off while the load is being lowered, the moment of the de- 
scending load plus that due to the rotating parts tends to lock 
the brake. In locking the brake, the external moment just 
mentioned must overcome the frictional resistance of the helical 
jaws and that between the discs g and n. The magnitude of 

the internal frictional moments is ^ (5 /jlD 4- d tan (a + <p') ) 

Letting Mi denote the moment due to the rotating parts and the 
inertia of the load, we obtain the following expression for the 
external moment: 



486 ANALYSIS OF THE SHAW BRAKE [Chap. XVII 

^ + Mi > ^ [5 pD + d tan (a + *>')] , 

from which (Q) - , n L " , 2 -7T (505) 

5 /*D + d tan (a + <p') 

The thrust (Q) becomes a minimum when the load comes to 

rest slowly, or' in other words, when the inertia forces become 

small and their effect may be neglected. Making M 1 = in 

(505), the minimum value of (Q) is given by the following 

expression: . 

iQl) = n(5/d) + d tan (a + *>')) (506) 

For all practical purposes, we may assume that (506) gives 
the magnitude of the axial thrust upon the helical jaws during the 
lowering period. 

The thrust (Q) becomes a maximum when the motor stops 
suddenly. The magnitude of (Q) for this case is given by (505) , 
in which 

= WRjM Ing) 
gn R 

(d) Condition of self-locking for lowering. — Assuming that the 
brake is to be self-locking for all loads, the most unfavorable 
condition arises when the axial thrust is a minimum, as given by 
(506) . The resistances that actually hold the load from running 
down, assuming the brake as self-locking, are those upon the 
discs /, g, m, and n. Equating the external moment, due to the 
load W, to the frictional moment of the discs, we have 

^ < 5 pDiQO (508) 

Combining (506) and (508), 

d tan (a + </) < 5 »D (509) 

The relation given by (509) must be fulfilled if the brake is 

to be self-locking during the lowering period. By comparing 

(504) and (509), it follows that if the latter is fulfilled, the former 

is also satisfied. 

(e) Moment required to release the brake.— Again assuming that 
the brake is self-locking, the motor must release the brake in 
order that the load may descend by gravity. The moment 
(M) required to release the brake must exceed by a small amount 
the sum of the frictional resistance of the helical jaws and that 
on the discs / and m, or 



Art. 335] DISPOSAL OF HEAT 487 

(M) > jI^hD + dtan (a - <?')] (510) 

The moment (M) becomes a maximum when Q' is maximum, 
which occurs directly after hoisting the load. To determine this 
maximum value of Q' , make a = in (501) and we obtain the 
relation expressed by (456). Substituting (503) in (510), the 
following expression for (M) is obtained: 

(M \ _WRr 5y,D + d tan (a - gQ n 

W nr] L5/J> + d tan (« + *>') J l j 

(/) Design constants and coefficients. — For design purposes, 
the coefficient of friction n for various combinations of materials 
may be assumed as follows : 

Wood against cast iron-/z varies from 0.25 to 0.35. 

Cast iron against cast iron lubricated-^ varies from 0.08 to 0.12. 

Cast iron against bronze lubricated-^ varies from 0.06 to 0.10. 

Cast iron against fiber lubricated-^ varies from 0.10 to 0.20. 

For screws and helical jaws that are well lubricated, the angle 
of friction <p f may be assumed as 5 degrees. 

The angle a varies from 5 to 17 degrees. 

The axial thrust per square inch of projected disc area varies 
between rather wide limits. An analysis of twelve brakes of 
various capacities showed that this pressure varied from 17 to 
270 pounds per square inch of disc area. 

The axial thrust per square inch of projected area of the screw 
thread or helical jaw for the above-mentioned twelve brakes 
varied from 90 to 1,800 pounds. 

335. Disposal of Heat. — The frictional resistance produced by 
a brake generates a certain amount of heat which is equivalent 
to the energy absorbed by the brake. Due to this fact, the 
brake should be designed so that the heat generated may be 
easily dissipated by conduction and radiation. Unfortunately, 
many brakes prove troublesome for the simple reason that 
the heat generated is not dissipated readily. 

The rise in the temperature of the brake sheave depends upon 
the amount of energy the brake is required to absorb every time 
it is applied and upon the frequency with which the brake is 
applied, as well as upon the weight of the rim and the specific 
heat of the material. In general, the effect of the arms and hub 
of the brake sheave is neglected in calculating the rise in 
temperature for a given case. 



488 DISPOSAL OF HEAT [Chap. XVII 

Prof. Nichols, in his "Laboratory Manual of Physics," gives 
the following formula for determining the rise in temperature 
due to radiation: 
Let A = the area of the radiating surface in square inches. 

T = the number of minutes the brake is at rest. 

c = the mechanical equivalent of the specific heat. 

k = the radiation factor. 

h = the lower temperature of the brake sheave. 

U = the higher temperature of the brake sheave. 

w = the weight of the brake sheave rim. 

Then 

. • . 0.434 kA T 

log (* 2 - h) = — (512) 

The energy absorbed by the rim is cw (t 2 — h), and equating 
this to the energy given up by the load and the rotating parts 
as given by (475), we obtain the following expression: 

E = cw (t 2 - h) (513) 

By means of (512), the approximate rise in the temperature of 
the brake may be determined, provided the factor k is known. 
Mr. E. R. Douglas, in an article entitled "The Theory and Design 
of Mechanical Brakes," published in the American Machinist 
of Dec. 19 and 26, 1901, states that "k generally lies between 
0.4 and 0.8 of a foot-pound of energy per minute for each square 
inch of surface and each degree Fahrenheit which that surface 
is above the temperature of the surrounding air." The lower 
temperature t x of the brake sheave may be assumed to vary from 
90° to 110°, while the temperature t 2 should not exceed 140° 
to 200°, depending upon the material forming the contact sur- 
faces. In order to prevent charring of the wood blocks or leather 
and fiber facings, the temperature t 2 should not exceed 150°. 

References 

Die Hebezeuge, by A. Ernst. 

Die Hebezeuge, by H. Bethmann. 

Machine Design, by H. D. Hess. 

Magnetic Brakes, Amer. Mach., vol. 25, p. 523. 

Brakes and Brake Mechanism, Machinery Reference Series, No. 47. 

Load Brakes, Amer. Mach., Aug. 20, 1903. 

Principles of Band Brake Design, Mchy., vol. 20, p. 386. 

Brakes, Mchy., vol. 12, p. 619; vol. 13, pp. 5, 61 and 117. 



CHAPTER XVIII 
SHAFTING 

336. Materials. — Shafts for practically all classes of service are 
subjected to shocks and jars. During each revolution the stresses 
in a shaft change from a maximum tension to maximum compres- 
sion, provided the rotating shaft is subjected to cross-bending. 
It is evident that the material for shafting must be tough and 
ductile. The common materials used for shafting are as follows. 

(a) Wrought iron. — In the past, engineers considered a good 
grade of wrought iron as the only material suitable for making 
shafts; but at present, due to its excessive cost of manufacture, 
wrought iron is used only in exceptional cases. Its strength is 
not as high as that of the modern steel that displaced it. 

(b) Bessemer steel. — Machinery steel made by the Bessemer 
process is used quite extensively for certain classes of machine 
shafting. It is cheap, and modern methods of manufacture give 
it sufficient ductility and toughness in the "mild grades" so that 
it is suitable for making shafts. One disadvantage of Bessemer 
steel is that it may contain hidden flaws or defects, though this 
is not a very common occurrence; hence Bessemer steel will ful- 
fill all the ordinary requirements in a large number of cases. 

(c) Open-hearth steel. — Steel made by the open-hearth process 
is more reliable in that it is more uniform than the Bessemer steel, 
and for this reason open-hearth steel is specified for many machine 
parts, such as armature shafts, engine shafts, shafts and spindles 
of machine tools, etc. 

(d) Alloy steels. — Many of the special steels described in Chap- 
ter II are used for making shafts for all classes of service, espe- 
cially when great strength is desired. Attention is again directed 
to the fact that shafts made from alloy steels possess no greater 
rigidity than the same size of shaft made of ordinary machinery 
steel. The shafts used on motor cars are made of high-grade 
alloy steels. The main shafts of marine and large hoisting 
engines are usually made of a high-grade nickel steel. In gen- 
eral, shafts made from alloy steels are more expensive than those 
made from common grades of steel. 

489 



490 COMMERCIAL SHAFT SIZES [Chap. XVIII 

337. Method of Manufacture. — Commercial shafting may be 
classified into the following groups: (a) Turned; (6) cold-rolled 
or drawn. 

(a) Turned shafting. — The ingot of steel, while hot, is rolled 
into bar stock having a diameter 3^6 i nc h greater than required 
for the finished shaft. The bar is then turned down in a lathe 
and polished accurately to size. It is evident that the diameter 
of the turned shafting is always }{& inch less than the so-called 
"nominal diameter." Large shafts are forged from an ingot, 
and turned down and finished accurately in a lathe. 

(6) Cold-rolled or drawn shafting. — To produce cold-rolled 
shafting, hot-rolled bar stock, previously treated with an acid so 
as to clean the outer skin, is passed through special rolls under 
great pressure, or drawn through special dies. This cold-rolling 
or drawing process renders the shaft fairly uniform in size. The 
surface acquires a polished appearance and becomes hard and 
tough. Experiments on cold-rolled and drawn shafting show 
that the strength of the material is increased, but at the same time 
the ductility is reduced. 

A disadvantage of cold-rolling or drawing lies in the fact that 
a considerable amount of skin tension is induced in the material 
of the shaft. This tension is relieved when a key seat is cut, 
thus causing the shaft to warp, and it must be trued up before 
it can be used. It is quite evident, therefore, that neither cold- 
rolled nor cold-drawn shafting is well adapted for use in high- 
grade machinery where accuracy is desirable. However, for the 
cheaper grades of machinery such shafts are used extensively. 

338. Commercial Sizes of Shafting. — Formerly, when wrought 
iron was used for shafting, the stock sizes of the hot-rolled bars 
from which the shafts were made varied by 3^-inch increments. 
Since these bars were reduced Y\% inch in finishing, the commer- 
cial sizes of shafts thus established varied by ^-inch increments 
but were always }>{$ inch less than each even y± inch in di- 
ameter. Later on, when steel replaced wrought iron, the list 
of stock sizes was increased. According to some of the prominent 
manufacturers of power-transmission machinery, it is possible to 
obtain turned shafting in the following sizes : 

From Y± to 2 inches, the diameters vary by }{Q-mch increments. 

From 2 to 6 inches, the diameters vary by ^-inch increments. 
Sizes which are }{q inch less than the even }i inch in diameter 
are also obtainable. 



Art. 339] SIMPLE BENDING 491 

Shafts larger than 6 inches in diameter are usually forged to 
order. 

All of the sizes which are K6 inch under the even l /± inch in 
diameter are generally accepted as standard for such appurte- 
nances as couplings, hangers, pillow blocks, etc. 

Cold-rolled or drawn steel shafting may be obtained in sizes 
from 3{ 6 inch and up, the diameters varying by Vi6-i nc h 
increments. 

SHAFT CALCULATIONS 

The straining actions to which shafting may be subjected are 
as follows: (a) simple bending; (6) simple twisting; (c) combined 
twisting and bending; (d) combined twisting and compression. 

339. Simple Bending. — In many classes of machinery, shafts 
are used that transmit no torsional moment, but merely support 
certain machine parts. Such shafts may revolve or remain 
stationary. In the latter case, the rotating machine parts sup- 
ported by the shaft are generally bronze-bushed or mounted on 
ball or roller bearings. The hoisting drum shown in Fig. 284 
is supported by a stationary shaft which is held rigidly by the 
supporting pedestals A and B. In the common car axle we have 
a good illustration of a rotating shaft subjected to a bending 
moment. 

(a) Strength. — The diameter of a shaft subjected to simple 
bending may be determined by equating the external moment M 
to the moment of resistance of the shaft. Thus 

32 ' 
from which 

~ s -t- <«« 

To facilitate making calculations, the second member of (514) 
may be evaluated for various diameters and the results arranged 
in chart form as shown in Figs. 281 and 282. The determination 
of the magnitude of the bending moment M depends upon the 
number of bearings supporting the shaft, and the distribution of 
the forces coming upon the shaft. The method of procedure in 
any given case is similar to that used in the case of beams. The 
value of the permissible fiber stress S varies from 5,000 to 35,000 
pounds per square inch and depends upon the material used for 
making the shaft. 



492 



STIFFNESS OF SHAFTING 



[Chap. XVIII 



(6) Stiffness. — In many machines the question of the stiffness 
of a shaft is of greater importance than that of its strength. In 
other words, for a shaft subjected to bending only the transverse 
deflection may have to be limited. These deflections depend 
upon the method of supporting the shaft as well as the distribu- 
tion of the forces acting on the shaft. To calculate the deflec- 
tions in a given case the formulas used in connection with beams 
will apply. No definite values are available for the transverse 
deflections of machine shafts, as they depend upon the service for 




\l l| 2 Zi l{ 2| 

Diameter d in inches 
Fig. 281. 



which the machine is intended. For line- and counter-shafts, a 
transverse deflection of 0.01 of an inch per foot of length is con- 
sidered good practice. 

340. Simple Twisting. — Shafting is very rarely subjected to 
simple twisting, since the weights of pulleys and gears, belt and 
chain pulls, and gear tooth pressures cause bending stresses. 
Frequently such bending stresses are difficult to determine before- 
hand, and due to the fact that the calculations become more or 
less complicated, many designers omit them in calculating the 
diameter of shafts. To make allowances for such unknown bend- 



Art. 340] SIMPLE TWISTING 493 

ing moments that are omitted, a low fiber stress is generally used 
in establishing the shaft diameter. Such a method of procedure 
should seldom be used. 

(a) Strength. — A long line- or counter-shaft having the pulleys, 
gears, or sprockets located near the bearings is generally con- 
sidered as a shaft transmitting a simple torsional moment. 
Ordinarily in such a shaft, the belt and chain pulls are not exces- 
sive and the bending moment caused by them may be omitted 
in the calculations for the diameter of the shaft. Hence, equat- 
ing the torsional moment of the load to the moment of resistance, 
we have 

T ird s S s 
16 ' 
from which 

T - - ^ (515) 

S. ~ 16' (515) 

The graphs of Figs. 281 and 282 may be found convenient in 

the solution of problems involving the use of (515), but it should 

M T 
be remembered that for the same diameter of shaft -5- = ^~cr • 

The magnitude of the permissible shearing stress S s varies from 
2,500 up. 

Substituting in (515) the value of T expressed in terms of the 
horse power transmitted and the revolutions per minute of the 
shaft, we obtain the following expression for the diameter of the 
shaft : 



4 



321,000 H ..... 

W— (516) 



in which H and N denote the horse power and revolutions per 
minute, respectively. According to the formulas recommended 
by several prominent manufacturers of power transmission ma- 
chinery, the shearing stress S e may be assigned the following 
values : 

1 . For well-supported head shafts carrying main driving pulleys, 
• sheaves, or gears and transmitting heavy loads, S s is approxi- 
mately 2,600. 

2. For regular line shafts supported on bearings every 8 feet, 
S s = 4,300. 

3. For light duty line shafts supported on bearings every 8 
or 10 feet, S s = 6,400. 



494 



CHART FOR SHAFTS 



[Chap. XVIII 



20- 



« 12 

lis 



i 



p 



40 



60 



80 



100 






120 ° 



140 



160 



180 



5 6 7 6 9 

Diameter d in Inches 
Fig. 282. 



10 II 12 



Art. 341] COMBINED TWISTING AND BENDING 495 

(6) Stifffiess. — In machine tools it is necessary that the im- 
portant drive shafts be made stiff so that they will not "wind up " 
like a spring. Such angular deflection must be limited in machine 
tools, while in other classes of machinery it need not be considered 
at all. To determine the relation between the torsional moment 
T and the angular deflection 0, the following method may be 
used. 

Since the torsional modulus of elasticity E t represents the 
ratio of the unit stress to the unit deformation, we get 

Et = §*, (517) 

in which I and x denote the length of the shaft and the deflection 
measured on the surface of the shaft, respectively. Both x and 
I are measured in inches. 

The length of the arc x is w^., and substituting this value 

in (517), we obtain the following expression for the angular 
deflection : 

„ 360 IS, 



TTClEt 



(518) 



Substituting in (518) the value of S s obtained from (515), we 
have 

e = - imr (519) 

For ordinary shafts, it is common practice to limit the angle 
to 1 degree in a length of shaft equivalent to 20 diameters. 

341. Combined Twisting and Bending. — A rotating shaft 
carrying pulleys, sprockets, sheaves, and gears is subjected to 
both bending and twisting when used for the transmission of 
power. Calculating the diameter of the shaft by means of either 
of the formulas (514) or (515), and ignoring the other, would 
result in a weak shaft. In designing shafts subjected to com- 
bined bending and torsion, several formulas based upon different 
theories are advocated by various investigators. The theories 
upon which these formulas are based are as follows: (a) the 
maximum normal stress theory; (b) the maximum strain theory; 
(c) the maximum shear theory. 

(a) Maximum normal stress theory. — The maximum normal 
stress or Rankine's theory is based upon the assumption that the 



496 MAXIMUM STRAIN THEORY [Chap. XVIII 

yield point depends upon the maximum normal stress, and not 
upon the shear or other stresses acting at right angles to it. The 
resulting maximum stress is calculated by the following formula : 



Max. normal stress S" = -+ Jsl + ~ (520) 

To facilitate the use of (520) when designing shafts, it has been 
found convenient to employ what is generally called the "equiva- 
lent twisting moment" T", an expression for which may be 
derived as follows: Substituting in (520) the values of S and S s 
in terms of the diameter d, we obtain 

Se = ^ (M + VM^TT 2 ), (521) 



from which 



T" = ' K ^- = M + \/M* + T 2 (522) 

Id 



The so-called equivalent twisting moment T'J will produce the 
same maximum normal stress as is produced by the combined 
action of M and T. In using (522), it is important to remember 
that S" is a tensile or compressive stress, and not a shearing 
stress. 

Some designers prefer to use an expression for the "equivalent 
bending moment" M" in place of (522). Multiplying and 
dividing (521) by 2, we obtain the following expression: 

from which 

M'J = ^f- =\{M + Vl^+T" 2 ) (523) 

The equivalent bending moment M" will produce the same 
maximum normal stress as M and T acting together. The 
allowable stress S'J must be the same as that used with (522). 

(6) Maximum strain theory. — The maximum strain theory, 
generally credited to Saint- Venant, is based upon the assump- 
tion that yielding of the material will not occur until a certain 
deformation has been produced. To determine the stress that 
produces yielding according to this theory, the following formula 
must be used : . 

Max. normal stress S e = (1 - m) | + (1+ m) ■yjsl + f' (524) 



Art. 341] MAXIMUM SHEAR THEORY 497 

in which the symbol m denotes Poisson's ratio, values of which 
are given in Table 1. For steel, m may be assumed as 0.3. 
Substituting this value in (524), and introducing the values of 
S and S s in terms of d, we get an expression for the equivalent 
twisting moment T c , as follows: 

T e = ^- e = 0.70 M + 1.3 \/M T +T 2 (525) 

The equivalent bending moment M e becomes 

M e = ~- r = 0.35 M + 0.65 y/W+T* (526) 

To decrease the numerical work involved in applying (525) 

M 
or (526) to any particular problem, let -^ = k; then (525) and 

(526) may be written 

T e = ^ = T(0.7fc + 1.3 VfcM 7 !) (527) 

and 

M e = T(0.35 k + 0.65 Vk 2 + f) - (528) 

To find the diameter of a shaft suitable for the combined 
moments M and T, substitute the value of T e for T and S e for 
S s in (515), and use the graphs of Figs. 281 and 282 as directed 
in Art. 340(a). If (528) is preferred, substitute the value of 
M e for M and S e for S in (514), and consult the graphs of Figs. 
281 and 282 for the diameter of the shaft corresponding to the 

M e 

calculated ratio -w-. 

(c) Maximum shear theory. — Up to the year 1900, the two 
theories just discussed were the only ones in use; at that time Prof. 
Guest reported in the Philosophical Magazine the results of his 
investigations upon the behavior of ductile materials subjected 
to combined stresses. His conclusion was that the yield point 
depends upon the maximum shearing stress ; that is, the material 
yields when the greatest resultant shear reaches a certain limit. 
The formula for calculating the maximum shearing stress that 
produces yielding is as follows: 



Max. shear S' e = As*. + — 



(529) 



To determine the equivalent twisting moment T e for this 
theory, substitute in (529) the values of S and S s ; hence 



498 



SELECTION OF PROPER THEORY [Chap. XVIII 



rd jf = VlPTT> 



= T Vfc 2 + 1 (530) 

To determine the diameter of the shaft by means of (530) in 

any particular problem, substitute T' e for T and S' e for S, 

in (515), and consult the graphs of Figs. 281 and 282 as directed 

in Art. 340(a). 

342. Method of Application.— In order to determine the 
diameter of a shaft subjected to combined twisting and bending, 
we must decide which of the theories discussed in Art. 341 should 
be used. It should be noted that the maximum strain theory 



i.o- 

1.9 

i.a 

1.7 










































































Values of c 






^=R 






r 
















""? I i" 












rh 































2.0 2.5 

Values of 
'Fig. 283. 



3.0 

= M 

T 



is really nothing more than a refinement of the maximum normal 
stress theory, and for that reason is more accurate. 

Comparing (527) and (530), it is evident that for equal 
diameters d, the following relation must exist between the 
allowable stresses and k: 



0.7 fc + 1.3V/c 2 + 1 

Vfc 2 + i 



(531) 



The relation expressed by (531) may be represented graphic- 
ally as shown in Fig. 283. Every point on the curve represents 
simultaneous values of k and the ratio c for which (527) and 
(530) will give the same shaft diameter. It is evident that if 
a point represented by the coordinates c and k does not lie on the 
curve, one of these formulas will give a diameter of shaft which 



Art. 343] COMBINED TWISTING AND COMPRESSION 499 

is larger than that given by the other. The object of representing 
(531) by the graph of Fig. 283 is to show at a glance which 
formula or theory must be used in a given case in order to ob- 
tain the maximum shaft diameter. If the coordinates c and k 
locate the point below the curve, the maximum strain theory 
must be used; that is, use formula (527) or (528). If the point 
lies above the curve, the maximum shear theory or formula 
(530) must be used. 

According to C. A. M. Smith, an English investigator, the 
ratio of the working stresses for mild steel in tension and shear is 
practically 2 to 1 instead of 5 to 4, as usually quoted in text books. 
In Table 95 are given the fiber stresses at the elastic limit for 
tension and shear as determined by Prof. Hancock. 



Table 95- 


-Fiber Stresses at the 


Elastic Limit 


Material 


Tension - 


Shear 


Ratio ~j 




47,000 
76,500 


30,500 
38,000 


1.54 




2.01 









343. Combined Twisting and Compression. — Shafts subjected 
to a twisting moment combined with a compression are frequently 
met with in machinery. Among the most common examples of 
such shafts are those used for driving worm gearing and the pro- 
peller shafts of ships. Occasionally vertical shafts carrying 
heavy rotating parts are subjected to combined twisting and 
compression. However, in many cases of worm gearing the 
worm can be mounted so that very little nf the thrust comes 
upon the shaft proper. 

(a) Short shaft. — The first case to be discussed is one in which 
the part of the shaft subjected to compression is so short that 
it may be considered as a simple compression member so far as 
the action of the thrust is concerned. 

4P 

The intensity of compressive stress for a solid shaft is S c = — -r' 

ira 

in which P denotes the thrust. From (515), the intensity of 
shearing stress due to the twisting moment on the shaft is 

1 (\ *V 
S s = — 7v • The resultant maximum stress due to the combined 

action of S e and S s may be found by substituting the values of 
the latter in (520) ; whence 



500 COMBINED TWISTING AND COMPRESSION [Chap. XVIII 

Max. compressive stress =.~ \P + JP 2 + -^-^ (532) 
7TC6 2 [ \ d 2 J 

To determine the diameter d of a shaft having given the 
magnitudes of the thrust P, the torsional moment T, and the 
allowable compressive stress, assume a trial value for d somewhat 
larger than that required for the twisting moment alone and 
evaluate (532). If the calculated value of the stress does not 
come near the allowable maximum make a second calculation, 
and so on. 

(b) Long shaft. — The second case to be considered is that of a 
shaft in which the part subjected to a thrust is so long that it is 
liable to buckle; in other words the shaft must be considered as 
a long column. According to Art. 15, the mean intensity of 
permissible compressive stress in a long column having a circular 
cross-section and subjected to a thrust P is as follows: 

S ' c = riP = ' S e L (533) 

+ m 2 r 2 E 

Assuming that the coefficient of elasticity E has an average 
value of 30,000,000, and that n may be taken as unity, (533) 
reduces to the following form : 

S/ = ~ = ~ %r- (534) 

*d 2 S e L 

^ 18,500,000 d 2 

The stress calculated by (534) is the mean intensity of com- 
pressive stress which corresponds to a maximum compressive 
stress S c in the long shaft; hence a short shaft having the same 
diameter as the longer one is capable of withstanding a thrust 
P' which is greater than P in the ratio of S c to S' c . It is evident 
that the magnitude of the thrust P' is given by the following 
expression : 

P' =P~ (535) 

To determine the diameter of the shaft necessary to support 
the thrust P and twisting moment T, use (532) as before, but 
substitute therein for P the magnitude of P' as calculated by (535). 

344. Bending Moments. — In calculating the bending moments 
coming upon a shaft supported on the ordinary type of bearings, 
it seems reasonable to assume that the clearance between the 



Art. 345] 



CRANE DRUM SHAFT 



501 



bearings and the shaft will permit the latter to deflect up to the 
middle of the bearings. Therefore, in such cases the moment 
arms should be measured to the middle of the bearings, and a 
shaft designed upon this assumption will generally be of ample 
size so far as strength is concerned. 

Whenever a gear, flywheel, or other machine part is forced or 
shrunk upon a shaft, it is practically impossible for the shaft to 
fail at the center of the hub. However, the shaft may fail along 
a section near either end of the hub, since any bending of the 
shaft would tend to localize the crushing at those sections. Ac- 
cording to Mr. C. L. Griffin, the critical sections may be assumed 
to lie from }^ to 1 inch inside of the hub. 

The majority of machine designers assume the moment arms 
as extending to the center of hubs and bearings, probably because 




2W 

Fig. 284 



m ll 






















'■'■:■;■'.■/?& 


r" c ~ 








| 








£ ^ 




H 


1* 


s Hc' 




F j 




3 1 












1 










—f— 


• 




G 


W 


D V\ 







the method is simple and the results obtained are on the side of 
safety. 

SPECIAL PROBLEMS 

The problems discussed in the following articles will serve to 
illustrate the general method of procedure that may be used in 
calculating the bending moments coming upon a shaft and finally 
in determining the diameter of the shaft required in any given 



345. Crane Drum Shaft. — In Fig. 284 is shown a crane drum 
running loose on the stationary shaft. With this construction 
no twisting moment is transmitted through the shaft. The first 
step to be taken in calculating the diameter of the shaft is to 
determine the magnitude and location of the maximum bending 
moment coming upon the shaft due to the loading shown in Fig. 
284. 



502 CRANE DRUM SHAFT [Chap. XVIII 

Let D = the weight of the drain- 
er = the weight of the gear. 
T = the tooth thrust due to the driving pinion. 
W = the load on each hoisting rope. 

Taking moments about the center of the supporting pedestal 
B, we have for the horizontal load at A the following expression : 

A h =j-(L-f), (536) 

and for the vertical load at A, we have 

A ^G(L-t) + W( b + 2c)+D g (537) 

L 

Combining these loads in the usual manner, we obtain the 
following expression for the resultant pressure at A : 

A = VAl+A* (538) 

Taking moments about the center of the supporting pedestal 
A, the horizontal load coming upon pedestal B is 

Tt 
B h = j-, (539) 

and the vertical load at B is 

B, = Gt + W(t + a + b) + D(L- 9 ) 
L 

Hence, the resultant pressure at B is 

B = VBf+Bl (541) 

The bending moment at the center of the bearing C is Ae, and 
that at the center of the bearing D is Bf; whichever of these 
moments is the greater must be used in calculating the diameter 
of the shaft by means of (514). 

346. Shaft Supporting Two Normal Loads between the Bear- 
ings. — A shaft supported on two bearings and carrying two or 
more gears, sprockets, or pulleys is of common occurrence in 
machinery. In some cases the gears are located between the 
bearings as shown in Fig. 285, while in others they are arranged 
as shown in Fig. 288. Furthermore, the loads coming upon the 
gears or pulleys produce bending moments that are either co- 
planar or in planes inclined to each other. 

(a) Diameter of shaft required for strength. — It is desired to de- 



Akt. 346] 



SHAFT PROBLEMS 



503 



termine the diameter of the shaft shown in Fig. 285, assuming 
the horse power H is transmitted at N revolutions per minute. 
The first step to be taken in the solution of this problem is to 
determine, by means of the following formula, the torsional 
moment T transmitted by the shaft : 



T = 63,030 



(542) 



Knowing the torsional moment T, we may readily calculate the 
magnitudes of the effort P and the resistance W, since 



PR = Wr 



(543) 



Having determined the forces P and W, we may treat the shaft 
as a simple beam and determine the bending moments at impor- 
tant points along the shaft. Since P and W act in planes that are 
at right angles to each other, the problem may be simplified by 




constructing the bending moment diagram for each of these forces 
and later combining these diagrams in order to determine the 
maximum moment. In Fig. 285 the triangle AEB represents 
the bending moment diagram for the force P, and AFB repre- 
sents a similar diagram for the force W. In other words, at the 
point D the shaft is subjected to two non-coplanar bending mo- 
ments; the one due to P is represented by the vector DE and the 
other due to Q is represented by the vector DG. These bending 
moments are in planes at right angles to each other; hence the 
resultant moment at D is equal to the vector sum of DE and DG, 
or 



M D = \JDE + DG 



(544) 



504 SHAFT PROBLEMS [Chap. XVIII 

In a i similar manner the resultant bending moment at C. is the 
vector sum of CF and CH, or 

M c = \JcF 2 + ~CH (545) 

For the shaft shown in the figure, it is evident that the maxi- 
mum moment occurs under the pinion, namely, at the point C. 
Having determined the magnitude of the maximum bending 

M 
moment M , calculate the value of the ratio K = -^,. For the 

o 
particular material used in the shaft determine the ratio —7, and 

S e 

by means of the graph of Fig. 283 ascertain which formula must 
be used to calculate the diameter of the shaft. The permissible 
stress S e or S' e depends upon the nature of the transmission and 
the material, and ordinarily it may be assumed as from 20 to 40 
per cent, of the stress at the elastic limit. 

(b) Diameter of the shaft required for stiffness. — It is required to 
determine the deflections at various points of the shaft shown in 
Fig. 285. Either the analytical or the graphical method may be 
used for ascertaining the deflections, but since the loads coming 
upon the shaft are non-coplanar the former method will prove to 
be the simpler. From the theory of a simple beam supporting a 
load Q, the deflection A x of the beam at any poing Xi inches from 
the left-hand support is given by the following expression: 

The deflection A 2 of the beam at any point x 2 inches from the 
right-hand support may be calculated by a formula similar to 
(546), namely, 

A 2 = §^ (L> -xi- a*) (547) 

The symbols used in the above formulas have the following 
significance: L denotes the distance between the supports; a 
the distance from the left-hand support to the load; b the dis- 
tance from the right-hand support to the load. 

Since the shaft shown in Fig. 285 may be treated as a simple 
beam, the deflections due to the force P may be calculated by 
means of (546) and (547). The deflections due to the load W 
may be determined in the same manner. Using the length of the 
shaft as a base line, the values determined by (546) and (547) 



Art. 347] 



SHAFT PROBLEMS 



505 



may be plotted, thus giving the deflection curve for each load. 
To determine the resultant deflection of the shaft at any point, 
due to the combined effect of P and W, find the vector sum of the 
separate deflections corresponding to the point under considera- 
tion. If the resultant deflection is considered too great for the 
particular class of service, increase the diameter of the shaft. 

347. Shaft Supporting Two Normal Loads with One Bearing 
Between the Loads. — (a) Diameter of the shaft required for strength. 
— A shaft, carrying gears or sprockets, supported on two bearings 
as shown in Fig. 286 is frequently used in machinery. One of 
the gears is located outside of the bearing B, thus causing a bend- 
ing moment at the center of this bearing. The magnitude of the 
bending moment at any point along the shaft, due to the force 
P, may be obtained by measuring the ordinate between AD and 




the lines AE and ED. Thus the magnitude of the moment at 
B is represented by the vector BE. In a similar manner the 
bending moment at any point along the shaft, due to the force 
W, may be determined by scaling the ordinate between A B and 
the lines AF and FB. 

Since the coplanar forces P and W are located on opposite sides 
of the bearing B and act in the same direction, it is evident that 
the magnitude of the resultant bending moment at any point 
along the shaft is given by the ordinate between the lines AFBD 
and AED. Combining the maximum resultant bending moment 
with the torsional moment transmitted, the diameter of the shaft 
may readily be determined by the method outlined in Art. 342. 

(b) Diameter of the shaft required for stiffness. — Having deter- 
mined the diameter of the shaft for the consideration of strength, 
it should be investigated for stiffness. For the shaft shown in the 



506 SHAFT PROBLEMS [Chap. XVIII 

figure, the analytical method of determining the deflections will 
prove simpler than the graphical method. The formulas for the 
deflection of the shaft, due to the action of the force P, are as 
follows : 

For any point on the shaft at a distance xi to the right of the 
bearing B 

Px, 
Ai = q^j (3czi - x\ + 2 cL) (548) 

For a point between the bearings at a distance x 2 to the left of 
the bearing B 

| A 2 = -■fijjjJzV' -x 2 )(2L-x 2 ) (549) 

The minus sign in (549) indicates that the deflection of the shaft 
between the bearings is in the opposite direction to the deflection 
of that part of the shaft overhanging the bearing. 

The shaft deflections due to the force W are in a direction oppo- 
site to those caused by the force P; hence, the resultant deflection 
at any point is the difference between the deflections due to the 
loads W and P. The deflections between the bearings due to W 
may be calculated by means of (546) and (547), while those to 
the right of the bearing B are given by the following expression : 

4 = - Wi (L + a) (55o) 

These deflections may be represented graphically as suggested 
in the preceding article, thus showing at a glance the location of 
the maximum. If the maximum deflection exceeds the permissible 
value the shaft diameter must be increased. 

348. Shaft Supporting One Normal and One Inclined Load 
between the Bearings. — In Fig. 287 is shown a shaft supported 
on two bearings carrying a spur and bevel friction gear. Due 
to the normal pressure P n between the contact surfaces of the 
friction gears, the shaft is subjected to an axial compression and 
bending moment in addition to the bending and torsional moments 
caused by the tangential forces on the two gears. The bending 
moments due to P, P n , and W may be calculated by the algebraic 
method or they may be determined graphically. Since P„ and 
W are coplanar forces, it is unnecessary to consider each of them 
separately in determining the bending moments. The force and 
funicular polygons shown in Fig. 287(6) and (c) are obtained in 



Art. 348] 



SHAFT PROBLEMS 



507 



the usual manner. Drawing the dividing ray OD parallel to the 
closing string od, we obtain in the vectors CD and DA the reac- 
tions at the bearings B and A due to the combined action of 
P n and W. The force P n represented by BF may be resolved 
into two components BC and CF as shown in the force polygon. 
The component CF produces a compression in the shaft and a 
thrust upon the bearing A. The magnitude of the component 
BC must equal the difference between the magnitudes of the 
pressures BE and EC produced upon the shaft at or near the 




Fig. 287. 



ends of the hub of the bevel friction gear by the inclined force 
P n . For all practical purposes the lines of action of the pressures 
BE and EC may be assumed as shown in Fig. 287(a) The 
magnitudes of BE and EC may be determined as follows : Pro- 
duce the string ob until it intersects the line of action be, and join 
this intersection with that of the string oc and the line of action 
ec; through the pole draw the ray OE parallel to the string oe, 
thus establishing the magnitudes of BE and EC. 

To determine the bending moment at any section of the shaft, 
as for example, along the line of action of the pressure BE, 



508 



SHAFT PROBLEMS 



[Chap. XVIII 



multiply the ordinate xy of the funicular polygon by the pole 
distance h. It should be remembered that the ordinate xy 
must be measured to the scale of the space diagram, while the 
pole distance h represents a force and hence must be measured 
to the scale of the force diagram. 

The tangential force P causes bending moments which are at 
right angles to those caused by the force W and P n ; hence, the 
method given in Art. 346 must be used to determine the maxi- 
mum resultant moment coming upon the shaft. Instead of 
finding the bending moments due to the force P by means of the 
graphical method, less labor is involved by using the algebraic 
method. If the shaft is long relative to the diameter, it is neces- 
sary either to treat it as a long column or to change the location 
of the bearing B. In other words, locating the bearing B adjacent 




Fig. 288. 



to the back of the bevel friction relieves the shaft from all column 
action, since the axial component would be absorbed by the bear- 
ing. Such an arrangement of bearings would in general be pre- 
ferred to that shown in Fig. 287(a). However, with the sug- 
gested change of bearings a force analysis different from that 
given above would be necessary. 

349. Two-bearing Shaft Supporting Three Loads. — Frequently 
shafts supported on two bearings and carrying more than two 
gears, sheaves, or sprockets are required. Such a shaft support- 
ing three loads is shown in Fig. 288. The bending moment 
diagram due to the load P on the large driving gear is represented 
by the triangle AEB, while that due to the loads W acting on 
the overhanging pinions is represented by CFGD. Since P 
and W are non-coplanar, the method of procedure for determin- 
ing the required diameter of the shaft is similar to that given in 



Art. 350] HOLLOW SHAFTS 509 

Art. 346. Having calculated the shaft diameter necessary for 
strength, the deflection must be investigated. For the shaft 
under consideration the deflections at several points along the 
shaft are readily obtained by the algebraic method, after which 
the deflection curves for the two systems of loads may be plotted. 
The maximum resultant deflection at any point may be de- 
termined from the curves, and if this is found excessive the 
diameter of the shaft must be increased. 

The deflection at any point between the overhanging load and 
the adjacent bearing, due to the action of the two equal loads 
W, may be calculated by the expression 

AV = ~-} (3 a(L + x,) - xl), (551) 

in which x\ denotes the distance from the bearing to the point 
under discussion. For the part between the bearings, the de- 
flection at any point due to the loads W is 

A'/ = - y^(L - x 2 ), (552) 

in which x 2 denotes the distance from the bearing to the point 
considered. 

350. Hollow Shafts. — In any shaft the outer fibers of the 
material are more useful in resisting the bending and twisting 
than the fibers at or near the center; hence the material may 
be distributed more efficiently by making the shafts hollow. 
Furthermore, the weight of such a shaft is diminished in greater 
proportion than its strength. 

It is evident that the strength of a hollow shaft is equivalent 
to the strength of the solid shaft minus the strength of the 
shaft having a diameter equal to the diameter of the hole. In 
determining the strength of the shaft having a diameter equal 
to that of the hole, the fiber stress to be used must be that 
produced in the solid shaft at a point whose distance from the 
center is equal to the radius of the hole. Letting S s denote the 
shearing stress produced in the outer fiber of the shaft having a 

diameter d x , the stress produced at a distance -~ from the center 

of the shaft is -j- S s . 
di 



510 HOLLOW SHAFTS [Chap. XVIII 

(a) Torsional strength. — For a hollow shaft the relation between 
the twisting moment and the diameters of the shaft is 

T ~ ~W ~I6^ ~ 16di {dl ~ d,) (553) 

Denoting the ratio of d 2 to di by u, the expression for the large 
diameter of a hollow shaft becomes 



<*> - 4^5 . (554) 

(b) Torsional stiffness. — The angular deflection, in degrees, 
caused by a given torsional moment T may be calculated by 
means of the following formula, obtained from (518) and (553) 
by eliminating S s : 

584 IT 
6 " %&-$) (555) 

(c) Transverse strength. — Occasionally it may be desired to 
calculate the diameter of a hollow shaft subjected to a given bend- 
ing moment M. This may be done by the use of the following 
formula, which is obtained by equating M to the moment of 
resistance of the hollow shaft and solving for d\: 



I 10.2 T 

\£(1 - u 4 ) 



d * = ylsor 1 ^*) (556) 

(d) Hollow and solid shafts of equal strength. — It is required to 
determine the relation between the diameter of a solid shaft and 
that of a hollow shaft of the same strength. Equating the mo- 
ments of resistance to twisting for the two shafts, we obtain 

Tfd O s TVO s , ,4 ,4s 



<*! = -yr=, (557) 

v 1 — w 4 

For the same strength, a hollow shaft is much lighter than a 
solid one. The per cent, saved in weight is given by the following 
formula : 



[ d* - d\ + dU 



per cent, gain = * ^ 2 100 (558) 



Art. 351] EFFECT OF KEY-SEATS 511 

EFFECT OF KEY-SEATS ON SHAFTING 

The effect of a key-seat in a shaft is to decrease slightly both its 
strength and stiffness. In order to obtain some knowledge as to 
the extent of this change in strength and stiffness, Prof. H. F. 
Moore of the University of Illinois made a series of tests, the 
results of which were reported in' Bulletin No. 42, University of 
Illinois Experiment Station. The shafts used in these tests 
varied in diameter from iy± to 23^ inches inclusive. Both cold- 
rolled and turned shafts made of soft steel were tested. The 
key-seats cut into these shafts were of common proportions. 

351. Effect upon Strength. — According to the results obtained 
by Prof. Moore, the ultimate strength of a key-seated shaft is 
practically the same as the ultimate strength of the solid shaft. 
Furthermore, very, little difference was observed between the 
strength of shafts with short key-seats and of similar shafts hav- 
ing long key-seats. The tests, however, showed conclusively 
that a key-seat has a decided influence upon the elastic strength 
of the shaft. In order to put the results of these experiments 
into usable form, the so-called "efficiency of the shaft" was deter- 
mined for each size of shaft tested. By the term "efficiency" is 
meant the ratio of the elastic strength of the shaft with the key- 
seat to the elastic strength of the solid shaft. The following 
equation for the efficiency is suggested by Prof. Moore as repre- 
senting fairly well the results he obtained : 

Ei = 1.0 - 0.2 w - 1.1 h, (559) 

in which w denotes the ratio of the width of the key-seat to the 
shaft diameter, and h, the ratio of the depth of the key-seat to 
the diameter of the shaft. 

352. Effect upon Stiffness. — A number of tests were also made 
to determine the effect of key-seats upon the angular stiffness 
of shafts. The following equation for the ratio of the angle of 
twist of the key-seated shaft to the angle of twist of the solid 
shaft is due to Prof. Moore, and may serve as a guide in determin- 
ing the probable weakening effect the key-seat has upon the tor- 
sional stiffness of the shaft. 

E 2 = 1.0 + 0.4 w + 0.7 A (560) 



512 REFERENCE [Chap. XVIII 

References 

Manufacture of Cold Drawn Shafting, Amer. Mach., vol. 41, p. 89. 

Production of Small Hollow Shafting, Amer. Mach., vol. 41, p. 367. 

Machinery Shafting, Machinery Reference Series, No. 12. 

Heavy Duty Shafts with Two and Three Bearings, Mchy., vol. 20, p. 659. 

Torque of Propeller Shafting, London Eng'g., Apr. 12, 1907. 

Stresses and Deflections of Shafts, Amer. Mach., vol. 37, p. 1027, and vol. 
38, p. 10. 

Charts for Critical Speeds, Amer. Mach., vol. 40. p. 809. 

Critical Speeds of Shafts, Amer. Mach., vol. 45, p. 505. 

Critical Speeds of Rotors Resting on Two Bearings, Amer. Mach., vol. 
46, p. 97. 

Critical Speeds of Rotors Resting on Three Bearings, Amer. Mach., vol. 46, 
p. 193. 

Critical Speed Calculations, Jour. A. S. M. E., June, 1910. 

Centrifugal Whirling of Shafts, Trans. Royal Soc, vol. 185 A, pp. 279- 
360. 



CHAPTER XIX 
BEARINGS AND JOURNALS 

BEARINGS 

353. Types of Bearings. — Bearings may be divided into two 
general classes: (a) sliding; (6) rolling. 

(a) Sliding bearings. — The sliding bearings in common use in 
machinery are of three types. The first type, called right line 
bearing, is one in which the motion is parallel to the elements of 
the sliding surfaces. These sliding surfaces may be flat, as the 
guides on engine crossheads and the ways of large planers and 
milling machines, or they may be angular as the ways on small 
planers and lathes. Circular guides are also in use for the cross- 
heads of engines and spindles of boring and drilling machines. 

The second type of bearing, called a journal bearing consists of 
two machine parts that rotate relatively to each other. The 
part which is enclosed by and rubs against the other is called 
the journal, and the part which encloses the journal is called, 
the box or less specifically the bearing. In the more common 
form of journal bearings, the journal rotates inside of a fixed 
bearing. In some cases, as in a loose pulley or a hoisting drum, 
the journal is fixed and the bearing rotates, while in other cases 
both the journal and the bearing have a definite motion, as for 
example, a crankpin and its bearing in the connecting rod. 

The thrust bearing is the third type of sliding bearing. It is 
used to take the end thrust in bevel and worm gearing, or in 
general any force acting in the direction of the shaft axis. Thrust 
bearings are of two kinds : (1) The so-called step- or -pivot-bearing, 
which supports the weight of a vertical shaft and its attached 
parts. The shaft terminates in the bearing. (2) The collar 
thrust bearing, which is used on propeller shafts, spindles of drill 
presses, and shafts carrying bevel and worm gears. In such 
cases the shaft generally extends through and beyond the bearing. 

(6) Rolling bearings. — Rolling bearings include all bearings in 
which rolling elements are used for supporting the rotating mem- 
bers. This class of bearings is discussed in detail in Chapter XX. 

513 



514 BEARING MATERIALS [Chap. XIX 

JOURNAL BEARING CONSTRUCTION 

354. General Considerations. — In designing bearings the fol- 
lowing important points must be given due consideration. 

(a) The proper bearing material must be selected with respect 
to the load" coming upon the bearing and the material used for 
the journal. 

(6) Provision must be made for anchoring the bearing material 
to the bearing proper. 

(c) Provision must be made for taking up any wear that is 
liable to occur. 

(d) In many cases means must be provided for preserving the 
alignment of the bearings. 

(e) Proper clearance between the journal and its bearing must 
be provided. 

(/) Means must be provided for lubricating the bearing. 

(g) Bearings running at high speeds* and subjected to high 
pressures must be provided with some means of dissipating the 
heat that is generated by friction. 

(h) The dimensions of the bearing, that is, the diameter and 
the length, are fixed by the journal with which the bearing is to 
run. The proportions of the journal are determined from a con- 
sideration of strength, rigidity, rubbing speed, and the permissi- 
ble pressure per square inch of projected area. 

355. Selection of Bearing Materials. — As a rule bearings give 
the best service when the material of the bearing and that of the 
journal are unlike. No satisfactory explanation has ever been 
offered why unlike materials are better, but it is claimed that 
with like materials the frictional resistance and the wear are 
greater. However, there are exceptions, as under certain condi- 
tions hardened steel against hardened steel, and cast iron in con- 
tact with cast iron have given excellent service. Bearing surfaces 
are made of many different substances depending largely upon 
the class of service for which the bearing is intended. The fol- 
lowing is a list of some of the materials that are used for bearing 
surfaces: babbitt metal; various grades of bronzes; cast iron; 
mild, case-hardened, and tempered steel; wood; fiber graphite. 

The main requirements for a good bearing metal are the follow- 
ing: (1) It should possess sufficient strength to prevent squeezing 
out of the bearing when subjected to a load. (2) It should not 
heat rapidly and should have a high melting point. (3) It 



Art. 355] 



BEARING MATERIALS 



515 



should be able to resist abrasion but should not score the journal. 
(4) It should be uniform in texture and possess a low coefficient 
of friction. 

(a) Babbitt metal. — Babbitt metal is used more extensively 
than any other bearing metal. One reason is that the metal is 
easily melted in a common ladle and poured into the bearing. 
Babbitt bearings require an outer shell to which the metal is 
anchored. Generally the shell is made of cast iron although steel 
casting and bronze are sometimes used. Shells made of bronze 
have the advantage that in case the babbitt metal melts and 
runs out of the bearing, the journal will not be damaged so read- 



bearings and from % to ^ inch thick in large bearings. To pre- 
vent rotation of the babbitt lining, the shell must be provided 
with some form of anchor. These anchors may consist of dove- 



Fig. 289. 

tailed grooves as shown in Fig. 289, or cored or drilled holes into 
which the babbitt may flow when the bearing liner is cast. 

Babbitt metals having various degrees of hardness are in use. 
A so-called hard babbitt is suitable for bearings subjected to 
heavy pressure or severe shock, while a soft babbitt is better 
adapted to a light load and high speed. Babbitt metal is used 
for the main bearings of engines and air compressors, on steam 
turbines, centrifugal pumps and blowers, motors and generators, 
in wood-working machinery, in bearings for line- and counter- 
shaft, and in many machine bearings of the split type. For the 
compositions of several grades of babbitt metals see Art. 51, 
Chapter II. 

(b) Bronzes. — Next to babbitt metal, bronze is considered the 
most important bearing material. It is commonly used in the 
form of a one-piece bushing forced under pressure into the shell 
or framework of the bearing. Frequently the bushing is split 
into halves each of which is fastened by suitable means to a part 



516 PROVISIONS FOR LUBRICATION [Chap. XIX 

of the bearing shell. The thickness of these bushings varies 
with the diameter and the length of the journal. There are 
upon the market a large number of different kinds of bronzes, 
many of which are giving excellent service. For the composition 
and other information pertaining to several grades of commercial 
bronzes see Art. 48, Chapter II. 

(c) Cast iron. — Cast-iron bearings running with steel journals 
have met with considerable success and eminent engineers have 
advocated their use, claiming that the surface will in a short 
time wear to a glassy finish and run with very little friction. 
However, if for any reason lubrication fails and heating begins, 
the result is liable to be either serious injury or total destruction 
to both bearing and journal. Several machine-tool builders 
use cast-iron bearings that are constantly flooded with oil and 
they experience no bearing troubles. In general it may be said 
that cast-iron bearings will prove satisfactory when the pressure 
and speed coming upon the bearing are not excessive and where 
sufficient lubrication is insured. 

356. Provisions for Lubrication. — The object of any system of 
lubrication is to form and maintain a uniform film of oil between 
the journal and its bearing. By the term system of lubrication 
is meant the method used for bringing the lubricant to the 
bearing and its distribution in the bearing. To distribute the 
oil and assist in the formation of a uniform oil film, the bearing is 
generally provided with a series of oil grooves. These grooves 
should start at the point of supply and lead diagonally outward 
in the direction of rotation. For journals rotating in either 
direction, the bearing is provided with a symmetrical arrange- 
ment of grooves. To insure the formation of the oil film, the 
edges of the oil grooves must be bevelled or rounded off. The 
lubricant is delivered to the bearing or to the journal in various 
ways among which are the following : (1) Drop-feed lubrication; 
(2) wick lubrication; (3) saturated-pad lubrication; (4) chain or 
ring lubrication; (5) flooded lubrication; (6) forced lubrication; 
(7) grease lubrication. 

(a) Drop-feed lubrication. — The most common method of oiling 
a bearing is by means of the drop-feed method. In its simplest 
form it consists of an open hole in the bearing through which oil 
is introduced. In many cases the hole is tapped to receive a 
closed oil cup, thus preventing dirt and grit from entering the 
bearing. 



Art. 356] 



PROVISIONS FOR LUBRICATION 



517 



(b) Wick lubrication. — In bearings used on line- and counter- 
shafts and occasionally in machinery, the oil is transferred by 
capillary action from a small reservoir in the cap to the bearing 
surfaces by means of a wick as shown in Fig. 290. This method 
of lubrication is satisfactory when the bearing pressures and the 
speed are not excessive. 




(c) Saturated-pad lubrication. — An effective way of lubricating 
line- and countershaft bearings is by means of wooden blocks con- 
taining a series of saw-cuts through which the oil rises. The 
blocks, generally two in number, are located in the lower half of 
the bearing and are held in contact with the shaft by means of 
springs. The lower ends of these blocks project into the oil 




Fig. 291. 



reservoir, thus permitting the lubricant to rise from the reservoir 
to the shaft by means of capillary action. 

(d) Ring or chain lubrication. — Ring or chain lubrication is 
considered one of the best methods of supplying a bearing with 
oil. It is used on bearings for all classes of machinery. An 
application of a ring oiler to a line- and countershaft bearing is 
shown in Fig. 291, and in Fig. 296, 297, 300, 310 and 311 are 
shown various designs such as are used on machine tools, cen- 



518 PROVISIONS FOR LUBRICATION [Chap. XIX 

trifugal pumps, etc. The quantity of oil delivered to the bearing 
by a ring depends upon the size and speed of the ring and upon 
the viscosity of the oil. The diameter of the oil ring should be 
made approximately double the diameter of the shaft, and the 
ring may be made solid or split. The former construction is 
used for small bearings and the latter for larger bearings. An 
inspection of a considerable number of ring-oiling bearings used 
on line-shafts, electrical machinery, and centrifugal pumps seems 
to indicate that an oil ring cannot be expected to supply proper 
lubrication over a length of bearing exceeding approximately 
4 inches on each side of the ring. In electrical machinery the 
rings are usually made of brass or bronze in order to avoid mag- 
netic difficulties. In general the rings should be perfectly round, 
they should have no sharp corners, and they should be well 
balanced. 

On the main bearings of high-speed engines a form of bearing 
similar to the ring-oiling type is occasionally used, but in place of 
the ring a sash chain is used. 

(e) Flooded lubrication. — In flooded lubrication the oil is sup- 
plied to the bearing by means of a pump or from an overhead 
reservoir, but at practically no pressure. This system has been 
used to some extent on machine tools. 

(/) Forced lubrication. — In forced lubrication the oil is supplied 
to the bearing at a considerable pressure by means of a pump. 
Generally the oil pressure varies from 15 to 25 pounds per square 
inch; however, the pressure may run up to 600 pounds per square 
inch as in the case of the step bearing used on Curtis vertical 
steam turbines. 

(g) Grease lubrication. — Grease lubrication is well adapted for 
use on bearings subjected to heavy pressures and in which the 
speeds are relatively low. Grease is introduced into the bearing 
by any one of the various forms of grease cups obtainable on the 
market. 

Very few of the systems of lubrication discussed above produce 
a perfect oil film. According to Axel K. Pederson, analytical 
expert of the General Electric Co., the various systems given 
above may be arranged into the following three classes: 

1. Those systems which produce an imperfect oil film; for ex- 
ample, drop-feed, wick, and grease lubrication. 

2. Those systems which produce a semi-perfect oil film, for ex- 
ample, saturated-pad and ring or chain lubrication. 



Art. 357] 



ADJUSTMENTS FOR WEAR 



519 



3. Those systems which produce a perfect oil film; for example, 
flooded and forced lubrication. 

357. Adjustments for Wear. — (a) Split bearing. — In the major- 
ity of bearings some means of taking up wear must be provided. 
The adjustment for wear may be made in various ways, but the 





Fig. 292. 

most common method is by the use of a split bearing the parts 
of which are bolted together. The wear is taken up by simply 
removing some of the metal or paper shims and tightening the 
bolts in the bearing cap. In split bearings the line of division 




Fig. 293. 

should be made with an offset as shown in Fig. 292 and 293 for 
two reasons: (1) When made with an offset, the cap will pre- 
vent the bearing under pressure from springing together at the 
sides and gripping the shaft. (2) The offset will, to a certain 
extent, prevent the escape of the lubricant. 



520 



ADJUSTMENTS FOR WEAR 



[Chap. XIX 



(b) Four-part bearing. — The main bearings of steam and gas 
engines are generally of the four-part type similar to the design 
shown in Fig. 294. The babbitt-lined side shells are provided 
with adjusting wedges which extend the full length of the bearing. 
The bottom shell is also lined with babbitt metal and rests in a 
spherical seat in the engine frame, thus keeping the shaft in good 
alignment at all times. By raising the shaft sufficiently to relieve 
the bearing of its load, the bottom shell may be rolled out and 
inspected. 

As shown in Fig. 294, the bearing cap is of heavy construction 
and is not babbitted the entire length of the bearing, but merely 
for a short distance at each end. The cap is placed over the 

































































I ' 






I \ 


-v. 




jLzzzzZ 









Fig. 294. 











ph 




^ 


_J 


m 


^ 






i 



jaws of the engine frame with a driving fit. It is evident that 
a four-part bearing permits making adjustments for wear in a 
more nearly correct manner than is possible with a common split 
bearing; hence it is well adapted for installation where the line 
of action of the resultant bearing pressure changes with the rota- 
tion of the shaft. 

(c) Solid bearing. — No doubt the simplest form of bearing is 
that known as the solid type, designs of which are shown in Figs. 
295 to 298 inclusive. The solid bearing has no provision for 
taking up wear except by removing the worn-out bushing or 
liner and replacing it with a new one. The bronze bushed bear- 
ings shown in Figs. 295 and 296 have been used successfully on 
heavy machine tools. They have ample provisions for lubrica- 
tion, but none for wear except by renewal of the bushing. Such 
bushings are replaced very readily at a small cost. In Fig. 297 
is shown another design of ring oiling solid bearing consisting of a 



Art. 357] 



ADJUSTMENTS FOR WEAR 



521 



cast-iron shell lined with babbitt metal. This type of bearing 
has been found to give good service on the small and medium 
sizes of centrifugal pumps. The shell and brass oil ring are fitted 
into a suitable housing; the shell is held in place by a special 
headless screw projecting into the hole shown in the figure. 











— 


L— i 


p 


J 








% 










----- 










XX 
CVJ 




^ 


W/A 


f 










( 






.---. 




\ 








k 


W/////A 


fi 


P 


=» 






\ 






M 
















Fig. 295. ■ 

The design of a solid bearing shown in Fig. 298 is used in places 
where the pressure upon the bearing is always in the same direc- 
tion, as for example on the shaft used for supporting the overhead 
sheaves of an elevator. As shown, merely the lower half of the 






P 
1 




W^ 




'/I 




-1—4 


i ' ) 


o 


: k 


nu 















Fig. 296. 



bearing, which in this case takes the entire pressure, is lined with 
babbitt metal. The central part of the bearing shell is made 
spherical so that it will fit into the spherical seat in the pedestal, 
thus keeping the shaft in proper alignment. 

A design of a solid bearing used on the spindles of heavy milling 
machines made by the Ingersoll Milling Machine Co. is shown in 



522 



ADJUSTMENTS FOR WEAR 



[Chap. XIX 



Fig. 299. The conical journal of the spindle a is fitted with a 
bronze bushing b, the latter being forced in the sleeve g. Some- 
where near the middle of its length, the spindle has keyed to it a 
removal conical journal c. The latter fits into the conical bronze 
bushing /, which is forced into the sliding sleeve g. Practically- 
all of the wear comes upon the bearing b and may be taken up by 




means of the adjusting nut e and the special washer provided 
between the end of b and the enlarged head of the spindle. Due 
to the use of the conical bearing, the alignment of the spindle is 
not disturbed by an appreciable amount when an adjustment for 
wear is made. 

On the spindles of certain machine tools the bearings are made 
with a bronze bushing having a straight bore and is turned conical 





Fig. 298. 

on the outside as shown in Fig. 300. The bushing is threaded at 
each end and is provided with a slit extending through the entire 
length. It is evident that this bearing may readily be adjusted 
for wear by means of the adjusting nuts at the ends of the bearing. 
The oil ring and oil reservoir provided in the framework of the 
bearing insure proper lubrication of the bearing at all times. 



Art. o5"t 



ADJUSTMENTS FOR WEAR 



523 



(d) Connecting-rod bearings. — The bearings used on connecting 
rods differ somewhat from those discussed in preceding para- 
graphs. In Figs. 301 to 303 inclusive are shown three designs that 
have proven satisfactory. The first two are used on the crankpin 
end of the rod while the third is intended for the crosshead end, 




Fig. 299. 

although a similar design is frequently used for the crankpin 
end. In all three cases the adjustments for wear are made by 
means of a wedge and suitable cap screws. 

The design shown in Fig. 301 consists of two half bearings 





Fig. 300. 



around which a steel stirrup or strap is placed, the latter being 
fastened rigidly to the rod end by two through bolts. The ad- 
justing wedge with its screws is located between the strap and 
the front half of the bearing. Taking up wear by means of this 



524 



ADJUSTMENTS FOR WEAR [Chap. XIX 



& 



3E 



3E 




Fig. 301. 



3E 



e- 



3E 




Fig. 302. 



Akt. 357] 



ADJUSTMENTS FOR WEAR 



525 



wedge tends to shorten the rod, hence the bearing at the other end 
of the rod should be equipped with an adjustment which tends to 
counteract the former, thus maintaining a constant distance 
between the two bearings. For economy of material the two 
halves of the bearing are made of steel casting lined with babbitt 
metal. Sometimes brass is used in place of the steel casting. 

In Fig. 302 is shown an open rod end into which are fitted the 
two halves of the bearing; one of these halves is movable and 
the other is fastened rigidly to the rod by a through bolt. The 
adjusting wedge and screws are located between the back bearing 



JZ7~ 



J 



& 



<> 



3E 




Fig. 303. 



and the rod, thus the tendency is to lengthen the rod when the 
wear is taken up. As in the design shown in Fig. 301, the two 
halves of the bearing are made of steel casting lined with babbitt 
metal. 

The design shown in Fig. 303 is called a closed-rod end. The 
adjustments for wear are made in the same manner as in the 
preceding designs. It is evident from the figure that taking 
up wear tends to lengthen the connecting rod, hence this design 
would be a proper one to use in connection with that shown in 
Fig. 301 since the length of the rod would remain practically a 
constant. The two parts of the bearing used with the closed- 



526 



ADJUSTMENTS FOR ALIGNMENT [Chap. XIX 



rod end of Fig. 303 are generally made of bronze though occasion- 
ally babbitt-lined bearings are used. 

358. Adjustments for Alignment. — In addition to provisions 
for taking up wear, many bearings are provided with means for 
aligning the shaft. Bearings that are out of line tend to heat 
and produce wear. Some of the bearings discussed in Art. 
357 meet the provisions for alignment by having the bearing 
divided into parts that can be adjusted vertically or horizontally, 
while others are provided with spherical seats thus making them 
self-aligning. In many cases the bearing and its housing are 





Fig. 304. 

mounted on supports which permit the adjustments necessary 
to line up the shaft. The horizontal adjustment in such cases 
is generally provided for by elongating the holes through which 
the housing is bolted to the support. 

Hangers. — For lining up the bearings of line- and countershafts 
various forms of hangers are used. As shown in Figs. 289 to 291 
inclusive, line-shaft bearings are made in two parts each of which 
is provided with a spherical seat which fits into a corresponding 
seat on the sockets of the hangers. A design of a cast-iron single- 
brace ball and socket drop hanger shown is in Fig. 304. From 
this figure it is evident that the two-ball seated sockets provide 
the- vertical adjustment while the slotted holes in the supporting 



Art. 359] 



BEARING PRESSURES 



527 



flanges of the hanger take care of the horizontal adjustment. 
When greater rigidity is required than is furnished by the hanger 
shown in Fig. 304, a double-brace design similar to that repre- 
sented in Fig. 305 is used. However, the hanger shown in the 
latter figure is made entirely of pressed steel, the parts being 

riveted or bolted together as shown. . . 

Set screws are used for giving the 
desired adjustments. 



■0- 



^ 



33- 



DESIGN OF BEARINGS AND 
JOURNALS 

359. Bearing Pressures. — In order 
to maintain an oil film between the 
journal and its bearing, the pressure 
must not exceed the so-called critical 
pressure, by which is meant the 
limiting pressure at which a perfect 
film between the journal and the 
bearing is maintained. This pres- 
sure depends upon the speed of the 
journal, the viscosity of the oil, the 
temperature of the bearing, the close- 
ness of the fit between the journal 
and its bearing, and the degree of 
finish given to the surfaces in con- 
tact. As yet no test results are 
available to show the relation exist- 
ing between the pressure, viscosity, and temperature. According 
to H. F. Moore, the relation existing between the critical pres- 
sure Pc and the speed of the journal is given by the following 
formula : 




Fig. 305. 



p c = 7A7VV, 



(561) 



in which V denotes the peripheral speed of the journal in feet 
per minute. The Moore formula is based upon the results 
obtained from a series of experiments on a steel journal running 
on a white metal bearing. The pressure carried on the bearing 
varied from 10 to 80 pounds per square inch of projected area, 
and the speed did not exceed 140 feet per minute. 

The following formula for the permissible bearing pressure 
based on Stribeck's results is taken from Smith and Marx's 



528 



BEARING PRESSURES 



[Chap. XIX 



"Machine Design," and is recommended for use when the 
speed does not exceed 500 feet per minute: 

Vc = 10VT (562) 

Table 96. — Allowable Bearing Pressures 



Type of bearing 


Pressure p 


Punching and shearing 
machinery- 


Main journals 
Crankpins 


2,000-3,000 
5,000-8,000 


Engine crankpins 


High speed 

Low speed 

Locomotives 

Marine 

Air compressors, center crank 

Auto gas engines 


250-600 
850-1,500 
1,500-1,700 
400-600 
250-500 
350-450 


Engine crosshead pins 


High speed 
Low speed 
Locomotives 
Marine 

jJAir compressors, center crank 
Auto gas engine 


900-1,700 

1,000-1,800 

3,000-4,000 

1,000-1,500 

400-800 

800-1,000 


Engine main bearings 


High speed 

Low speed 

Marine 

Air compressors, center crank 

Auto gas engine 


180-240 
160-220 
200-400 
150-250 
350-400 


Locomotive driving journals 


Passenger 

Freight 

Switching 


190 
200 
220 


Engine crossheads 


Stationary 
Marine 


25-40 
50-100 




300-500 




40-80 




40-60 




80-100 


Hoisting machinery shafting 




70-90 






Propeller shaft thrust bear- 
ings. 


Freight steamer 
Passenger steamer 
Large naval vessels 
Light naval vessels 


40-55 

60-80 

70-90 

110-130 



Art. 360] RELATION BETWEEN LENGTH AND DIAMETER 529 

For speeds exceeding 500 feet per minute, the same authorities 
suggest the formula 

Vc = 30 VV (563) 

In L. P. Alford's book on "Bearings" is given a chart showing 
the relation between the maximum safe bearing pressure and the 
rubbing speed for perfect film lubrication. This chart represents 
the practice of the General Electric Co. in designing the bearing 
used on motors and generators. The following expression gives 
values of the maximum safe bearing pressure which agree very 
closely with those obtained from the chart. 



15.5\/¥ 



(564) 



In addition to the formulas given in this article, the allowable 
bearing pressures, in pounds per square inch of projected area, 
contained in Table 96 will serve as a guide in designing bearings 
and journals. These values are based upon current practice 
and were collected from various sources. 



Table 97. — Relation between Length and Diameter of Bearings 







Ratio l/d 






Ratio l/d 




Min. 


Max. 




Min. 


Max. 


Marine 
engine 


Main bearing 
Crankpin bearing 


1.00 
1.00 


1.50 
1.50 


Steam turbines 
Generators and motors 


2.0 
2.0 


3.0 
3.0 


High-speed 
engine 


Main bearing 
Crankpin bearing 
Crosshead pin bearing 


2.00 
1.00 
1.40 


3.00 
1.00 
1.60 


Centrifugal pumps 
Centrifugal fans 
Machine tools 


2.0 
2.0 
2.0 


2.5 
3.0 
4.0 


Slow-speed 
engine 


Main bearing 
Crankpin bearing 
Crosshead pin bearing 


1.75 
1.00 
1.20 


2.25 
1.25 
1.50 


Hoisting drums 

Hoisting sheaves for cranes 

Wood working machinery 


1.5 

1.0 
2.5 


2.0 
2.0 
4.0 


Stationary- 
gas engine 


Main bearing 
Crankpin bearing 
Crosshead pin bearing 


2.00 
1.00 

1.50 


2.50 
1.50 
1.75 


Shaft 
hangers 


Rigid 
Self-adjusting 


2.5 
3.0 


3.0 
4.0 




Pillow 
blocks 


Plain 
Ring-oiling 


2.5 
4.0 


3.5 
5.0 


Auto gas 
engine 


Main bearing 
Crankpin bearing 
Crosshead pin bearing 


1.00 
1.20 
1.70 


1.75 
1.40 
2.25 



360. Relation between Length and Diameter. — The ratio of 
the length of a bearing to its diameter is fairly well established 
for the different classes of machinery. In Table 97 are given the 
values of this ratio for a considerable number of different types 



530 RADIATING CAPACITY [Chap. XIX 

of bearings, the majority of which were obtained from a study 
of actual installations. 

361. Radiating Capacity of Bearings. — The capacity that a 
bearing has for radiating the heat generated by the friction be- 
tween the journal and its bearing depends upon the mass of 
metal used in the construction of the bearing and upon the 
condition of the surrounding air. The following formula due 
to Axel K. Pedersen may be used for determining the amount 
of heat carried away: 

Q = ( ^±m, (565) 

in which Q denotes the heat radiating capacity of a bearing ex- 
pressed in foot-pounds per second per square inch of projected 
area; T denotes the difference between the temperature of the 
bearing and that of the cooling medium; K denotes an experi- 
mental constant the magnitude of which depends upon the 
method used for cooling the bearing. The following values of 
K derived by Pedersen from Lasche's and the General Electric 
Co.'s experiments may be safely used in designing bearings: 

1. For bearings of light construction located in still air — ■ 
K = 3,300. 

2. For bearings of heavy construction and well ventilated — 
K = 1,860. 

3. For General Electric Co.'s well-ventilated bearings — • 
K = 1,150. 

362. Coefficient of Friction. — The coefficient of friction be- 
tween the bearing and its journal depends upon the bearing 
pressure, the speed of the journal, the temperature of the bear- 
ing, the specific gravity of the lubricant, and the method used 
for lubricating the bearing. The laws governing the coefficient 
of friction in a bearing provided with a limited supply of lubri- 
cant are generally assumed the same as those governing ordinary 
sliding' friction. However, when the bearing is provided with a 
copious supply of lubricant, the coefficient of friction depends 
upon the laws of friction in a fluid, that is, the resistance the 
lubricant offers against shearing. 

In an article entitled "Bearing Design Constants" which 
appeared in Power, Feb. 22, 1916, Mr. Louis Illmer gives several 
formulas for the coefficient of friction which are based upon the 



Art. 362] COEFFICIENT OF FRICTION 531 

experimental researches of Tower, Lasche, Thomas, Maurer 
and Kelso. The coefficient of friction according to the Tower 
tests is given by the expression 



2 jV 

?7W 



(566) 



in which p denotes the bearing pressure in pounds per square 
inch of projected area; V the speed of the journal in feet per 
minute; T the virtual temperature head of the oil, which may 
be assumed as the temperature of the bearing less 60°. Ac- 
cording to Illmer this formula is applicable to bearings having a 
pressure range of 100 to 500 pounds per square inch of projected 
area, and in which the speed does not exceed 500 feet per 
minute. 

The Lasche experiments were made on a steel journal running 
in a ring oiling babbitt lined bearing, and the results obtained 
lead to the following expression for the coefficient of friction: 

» = WT (567) 

This formula is applicable to bearings subjected to pressures 
of 15 to 225 pounds per square inch of projected area, and in 
which the speed may range from 500 to 3,500 feet per minute. 
The temperature of the bearing may vary from 85° to 210°F. 

From the results of experiments made by Thomas, Maurer and 
Kelso on babbitt-lined hanger bearings, Illmer derived the fol- 
lowing formula for the coefficient of friction : 

Vv 

(568) 



20VpT 

The use of this formula is limited to bearings in which the pres- 
sures vary from 33 to 100 pounds per square inch of projected 
area, and in which the speed of the journal ranges from 100 to 
300 feet per minute. 

The experiments of Lasche, as well as some made at Cornell 
University, seem to indicate that the coefficient of friction is 
practically independent of the speed when the latter exceeds 
500 feet per minute. Upon this assumption, (568) may be sim- 
plified by substituting for V the critical value 500, whence 

(569) 



532 DESIGN FORMULAS [Chap. XIX 

Equation (569) proposed by Illmer, gives values of the coefficient 
of friction that may reasonably be expected in the operation of 
well-designed bearings lined with babbitt metal and lubricated 
with a generous supply of mineral engine oil. 

Mr. William Knight, in the American Machinist of Nov. 16, 
1916, suggests that (569) be modified by introducing in the nu- 
merator a factor s denoting the specific gravity of the oil when 
compared to water. Thus the revised form of (569) becomes 

Knight bases his suggestion upon an investigation of the results 
obtained by A. L. Westcott from a series of tests made at the 
University of Missouri on greases and oils. Furthermore (570) 
gives values of n that agree fairly well with the results obtained 
by Lasche for pressures between 120 and 240 pounds per square 
inch. 

363. Design Formulas. — Having given the ratio between the 
length of the bearing and its diameter, we may readily develop 
working formulas for the diameter of the bearing in terms of the 
load P, the revolutions per minute, and certain constants. The 
resultant formulas will be based upon equations (562) and (563) ; 
hence they will only apply to bearings receiving a copious supply 
of lubricant and to those in which the speed remains within the 
range given in Art. 359. For a bearing having a diameter d and 
length Z and subjected to a total pressure P, the pressure per 
square inch of projected area is 

V = £• (571) 

in which c denotes the ratio of I to d. Equating the value of p 
to the limiting pressure given by (562), we obtain 

P = 10 cd V7 (572) 

Introducing the value of V in terms of d and N, the number of 
revolutions per minute of the journal, we obtain the following ex- 
pression for d for speeds below 500 feet per minute: 

d = 0M ^ (573) 

By a similar procedure, using (563) in place of (562), we obtain 



Akt. 364] TEMPERATURE OF BEARINGS 533 

the following formula for d for speeds exceeding 500 feet per 
minute: 

d = 0.282 lE (574) 



By means of (573) or (574), whichever applies to the problem 
under discussion, the diameter of the bearing may be calculated. 
Knowing d, the length of the bearing may be determined since 
I = cd. Furthermore, the magnitude of p may be determined by 
means of (571). 

364. Temperature of Bearings. — Frequently it is desirable to 
determine the probable temperature of the bearing due to the 
heat generated. If the temperature becomes too high the oil is 
liable to lose its lubricating qualities, hence it may be necessary 
to redesign the bearing or resort to artificial cooling. The work 
of friction expressed in foot-pounds per second per square inch 
of projected area is 

W,-*& (575) 

and this must equal the quantity of heat radiated or carried away 
as expressed by (565) ; hence 

upV (T + 33) 2 



60 K 



(576) 



from which the limiting speed of the bearing for a given final 
temperature is 

v = jrk (To + 33)2 (577) 

Equation (577) may also be used to calculate the probable 
temperature of a well-lubricated bearing running under given 
conditions of load and speed. To determine this temperature 
the following method of procedure is suggested: From (568) or 
(570), depending upon the speed, determine the value of ix in 
terms of T. Substitute this value of ju as well as the magnitudes 
of P, V, and K in (577) and determine the probable temperature 
of the bearing. The maximum temperature of a bearing depends 
upon the lubricant used, and since bearing oils begin to show signs 
of losing their lubricating qualities at a temperature of approxi- 
mately 250°F. it is considered good practice to limit the maximum 
temperature, as determined by (577), to 180°F. 



534 DESIGN OF JOURNALS [Chap. XIX 

365. Strength and Stiffness of Journals. — In the majority of 
cases the journal is integral with and forms a part of the shaft, 
the diameter of which has been calculated according to the 
methods given in Chapter XVIII. The dimensions of the 
journals of important shafts are not generally based on calcula- 
tions for strength and stiffness but on the liability of heating, 
that is, the conditions which govern the oil supply. However, the 
stresses in a journal should always be investigated in order to 
make sure that the dimensions are ample so far as strength and 
rigidity are concerned. 

(a) Strength of end journals. — End journals are generally con- 
sidered cantilever beams loaded uniformly. Equating the bend- 
ing moment to the moment of resistance and solving for the diam- 
eter d, we obtain 



1.72 



^ (578) 



Having given the dimensions of the end journal, and the load 
coming upon it, the magnitude of the stress may be determined 
by (578), or by means of the formula 

■£ = 5.1 pc 2 , (579) 

in which p and c have the same meaning as assigned to them in 
Art. 362. The working stress S, due to the fatigue of the mate- 
rial, should not exceed 4,000 to 5,000 pounds per square inch. 

(6) Stiffness of end journals. — In designing journals the ques- 
tion of stiffness is an important one, and should be given the 
proper consideration. For an end journal loaded uniformly, the 
deflection A is calculated by the formulas 

2.55 PI _ 2.55 pc* ( 

A " Ed* ~ Ed (580) 

whence 

For common end journals good engineering practice dictates that 
the value of A should not exceed 0.01 of an inch. 

366. Design of Bearing Caps and Bolts. — The cap of a bearing 
should never be subjected to a heavy load; however, there are 
cases in which the circumstances are such that a considerable 
pressure comes upon the cap. In such cases the cap is generally 



Art. 366] 



DESIGNS OF CAPS AND BOLTS 



535 



regarded as a beam supported by the holding down bolts or 
screws and loaded at the center, as shown in Fig. 306. As in 
the journal, the cap should be investigated for both strength and 
stiffness. 

(a) Strength of cap. — Assuming the dimensions of the cap as 
represented in Fig. 306, we obtain the following expression for 
the thickness 6, by equating the bending moment to the moment 
of resistance: 

b = Jf (582) 

(b) Stiffness of cap. — In order that the cap will have ample 
rigidity, the thickness b should be calculated by the following 

















! 














Fig. 306. 



expression based upon the formula for the deflection A of a simple 
beam loaded as shown in Fig. 306: 



b = 0.63 a 



IAE 



(583) 



For the cap of a common end journal or a marine end connecting 
rod, good engineering practice limits the deflection A to 0.01 of 
an inch. 

(c) Holding-down bolts. — The bolts, screws, or studs that are 
used for holding down the cap are generally assumed to be sub- 
jected to a simple tension, and as a rule each bolt is designed for 

4P 

a load equivalent to -5—, in which n denotes the number of bolts 

used for holding down the cap. 

367. Work Lost Due to the Friction on a Cylindrical Journal. 

— With our present state of knowledge of the subject of friction, 
we are unable to determine a correct expression for the work 
lost due to journal friction. In deriving an expression for the 
moment of journal friction, it is generally assumed that the coef- 
ficient of friction is constant for a given speed and further that 
the pressure between the surfaces in contact is uniformly dis- 



536 FRICTION LOSSES IN BEARINGS [Chap. XIX 

tributed, or that the wear of the journal and its bearing is uni- 
form and proportional to the work of friction. The assumption 
of uniform distribution of pressure is hardly warranted in the 
case of a "worn-in" journal and bearing; but for a new journal 
and bearing having perfect contact over the entire bearing sur- 
face, it is probable that the pressure is uniformly distributed. 
In the following analysis, formulas based on both assumptions 
will be derived. 

(a) Pressure uniformly distributed. — Assuming that the pres- 
sure between the journal and its bearing is uniformly distributed 
over the contact surface, the intensity of pressure p is equal to 
the load P on the journal divided by the projected area of the 
journal. This may be shown as follows: 

The pressure on a longitudinal strip of width ds and length I 
is plds. Let. the direction of the pressure p make an angle 
with the vertical center line of the journal, and assume that the 
load P acts in vertical direction; then the component of p parallel 
to the line of action of P is 

dP = plcosdds =^ cosddd, (584) 

from which 

P = pld, 

P » 

or v = U ^ 585 ) 

The force of friction on the elementary strip Ids is uplds, and 
the moment of this force about the axis of the journal is 

lipid , 



whence by integration 



dM 2 



M = ^ (586) 



The work, in foot-pounds, lost per minute due to the friction 
is given by the formula 

W, = *=^, (587) 

in which the diameter d is expressed in inches, and N denotes the 
revolutions per minute. 

(6) Uniform vertical wear. — The statement that the normal 
wear is proportional to the work of friction is equivalent to say- 
ing that the normal wear n is «qual to the product of a constant 



Art. 367] FRICTION LOSSES IN BEARINGS 537 

k, normal pressure p, and the diameter dof the journal, since work 
is proportional to the product of p and d. Hence 

n = kpd (588) 

It is evident that the normal wear of a journal and bearing 
is greatest at the bottom and becomes zero at the sides. If 
the journal and bearing remain cylindrical after being worn it 
is apparent that the vertical wear h is constant, and the normal 
wear at any point of the surface in contact will be given by the 
relation 

n = h cos0 (589) 

Combining (588) and (589) 

p = C cos0, (590) 

in which the constant C = r-v Substituting (590) in (584), 
we get 

dP = ™ cos 2 0d0, 

from which the total load upon the bearing is 

P = ~ (591) 

The moment of the force of friction about the axis of the journal 



whence 


am = — 
M ■■ 


—. — cosvav, 
4 ' 

»Cld 2 
2 




(592) 


Eliminating C by combining (591) and (592), we 


;get 






M = 


= ^Pd 

7T 




(593) 


The energy, 


in foot-pounds 


;, lost per minute 


is given 


by the 


expression 


W f 


»NPd 
3 




(594) 



368. Work Lost Due to the Friction on a Conical Journal. — 

The expressions for the moment of friction and the work lost 



538 



FRICTION LOSSES IN BEARINGS [Chap. XIX 



due to the friction on a conical journal, having the dimensions 
shown in Fig. 307, are determined in a manner similar to that 
used in Art. 367. 

(a) Pressure uniformly distributed. — On the assumption of 
uniform distribution of pressure, the vertical component of 
the normal pressure on an elementary area is given by the 
expression 

dP = -^V cosddddr (595) 

tan a 

from which 



tan a L J 



(596) 





, — 1 


M— - 


P 




I 








* 


8 




^0 






[ 
- — I - 


] , 


- 




Fig. 307. 

The force of friction on the elementary area is — > and 

J cos a. 

the moment of this force about the axis of the journal is 



dM 



whence 



M = 



upr 2 



fXTp 



drdd; 



3 sin 
juttP 



3 cos 



a [rl - r\\ 



The energy, in foot-pounds, lost per minute is 



W, 



WW 



18 cos 



>_ [ rl - rX \ 



(597) 



(598) 



(599) 



(b) Uniform vertical wear. — Assuming that the vertical wear 



Art. 368] FRICTION LOSSES IN BEARINGS 539 

h of the journal and bearing remains constant for all points, the 
normal wear at any point is 

n = h cos a cos 9 (600) 

Since the normal wear is proportional to the work of friction, 
it is evident that 

n = kpr (601) 

Combining (600) and (601), we get 

v = ^ (602) 

Jl COS a 

in which the constant C = — t Substituting the value of 

p in (595), we get 

dP = — — cos 2 6dddr; 
tan a 

whence the total load P upon the journal becomes 



2 tan 



G> - r-d (603) 



To determine an expression for the moment of friction, substi- 
tute (602) in (597) ; whence 

dM = -^- r cosddddr 
sm a 

Integrating 

¥-£;[*-*! (604) 

Combining (603) and (604) in order to eliminate C, the magni- 
tude of the moment of friction of the conical journal is given by 
the expression 

M = -2^, (605) 

ir COS a 

in which d denotes the mean diameter of the conical journal. 

To calculate the energy lost due to friction, the following 
formula may be used: 

W, = I*™ (606) 

3 cos a K ' 

369. Proportions of Journal Bearings. — In general the dimen- 
sions of the various parts of a bearing are determined by means 
of empirical formulas which are based upon the diameter of the 



540 BEARING PROPORTIONS [Chap. XIX 

shaft. Such formulas usually give a well-proportioned bearing 
having an excess of strength. 

(a) Common split bearings. — The empirical formulas given 
below are based upon a series of dimensions obtained from 
several sizes of common split bearings similar to the type repre- 
sented by Fig. 292. The cap is held down by either two or four 
bolts, studs or cap screws, the number depending upon the 
length of the bearing. In the following formulas the symbols d 
and I denote respectively the diameter of the shaft and the length 
of the bearing: 

Outside diameter of bearing = 1.75 d + 0.5" 
Span of bolts = 1.7 d + 0.7" 
Distance between bolts '= 0.5 I \ (607, 

Size of bolts = %& d + 0.25' 
Thickness of babbitt = >{ 6 d + 0.121 

(6) Pedestal bearings. — The pedestal bearing shown in Fig. 293 
is provided with removable bearing shells which are made alike 
so that they are reversible. The shells are lined with babbitt 
metal that is peened, then bored and scraped to exact size. This 
type of bearing is manufactured by the Stephens-Adamson Mfg. 
Co. in six sizes ranging from 3 1 %6 to 9^ inches in diameter. 
The empirical formulas given below were derived from dimen- 
sions furnished by the manufacturer, and the various symbols 
used in these formulas apply to the key drawing of Fig. 293. 

a = 3.7 d + 3" m = 1.47 d + 0.25 

b = 3.1 d-h 1.75" n = 1.5 d 

c = 3 d + 0.8" p = 0.88 d + 1.8" 

e = 1.9 d + 0.5" q = 0.7 d + 0.75" 

f = g + 1.5" r = 0.75 d - 0.5" 

g = 2 d s = 0.38 d + 0.5" 

h = 1.7 d - 0.3" t = 0.5 d + 0.1" 

k = 1.43 d + 1.3" u = 0.28 d + 0.4" 

Diam. of bolts for base = 0.14 d + 0.45" ] 

Diam. of bolts for cap = 0.2 d - 0.08" J- (609) 

Thickness of babbitt = 0.025 d + 0.18" J 

(c) Rigid post bearings. — In Fig. 308 is shown a form of bab- 
bitt-lined split bearing that is used for carrying line- or counter- 
shafts when it is necessary to support the latter from posts and 



(608) 



Art. 369] 



BEARING PROPORTIONS 



541 





p 

a 
5 


CI 


to to 




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to tO tO 

to XrH \rn \rH to to 
X?J Vh \oo i-N V* co\ \oo m\ x-h \oo vm V* 

r-Ko5\llJs,H COX. rH t-\ rH 1-H rH\ rH\ Co\ rHX 




to 


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to \H Xr- I to to V-l 
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OSX CO\ rH iH r-i\ rH\ CO\ t-\ >-i\ OSX W\ t-X rH 




ia 


to to to to 
p-\ co\ l(K \oo \oo y, \N \N V-i V* \a> «\ 

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5 
pq 


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iH COX t-\ rH rH COX l>X iH rH CO\ t-X rH rH 

HHHHNNNNMnnn 



542 



THRUST BEARINGS 



[Chap. XIX 



columns. Bearings for a similar service but furnished with oil 
rings are also obtainable. In Table 98 are given the dimensions 
pertaining to the different sizes of the type of rigid post bearing 
illustrated in Fig. 308. 

THRUST BEARING CONSTRUCTION 

The journal bearings discussed in the preceding articles are 
not suitable for supporting vertical shafts carrying heavy rotating 




parts or horizontal shafts subjected to heavy pressures acting in 
a direction parallel to the shaft. However, in many installations 
of horizontal shafts subjected to axial loads, as for example, 
thrusts due to bevel and worm gearing, the ordinary journal 
bearing is used and the axial pressures are taken care of by means 
of one or more loose washers located between the supporting 
bearing and a suitable collar on the shaft. 

In all thrust bearings the speed of the surfaces in contact is 
a maximum at the outer edge, and at the axis, theoretically, it is. 
zero. At any point of contact the wear is proportional to the 
work of friction; namely, the product of the pressure at the point 



Art. 371] 



COLLAR THRUST BEARINGS 



543 



and the velocity of the point. The exact distribution of the 
pressure existing between the contact surfaces is not known 
definitely, but very likely it is maximum at the center, and for 
that reason a well-designed flat pivot bearing should have ring 
contact. This form of contact surface is produced by merely 
removing some of the metal at the center. 

370. Solid Bearing with Thrust Washers. — A solid bearing 
provided with thrust washers and used for supporting a bevel- 
gear transmission is shown in Fig. 309. The thrust of the gear 
is taken up by a single bronze washer, while that of the pinion is 
taken up by three washers, the two outer ones being made of 
steel and the other of bronze. The steel washers have spherical 
faces which fit into the spherical seats furnished on the hub of the 




Fig. 309. 



pinion and the end of the bearing. Frequently bearings of this 
description are furnished with a casing for enclosing the gear and 
pinion thus permitting the gearing and washers to run in an oil 
bath. In addition to providing an effective means of lubrica- 
tion, the casing also protects the workmen from coming into con- 
tact with the gearing. The thrust due to worm gearing is fre- 
quently taken care of by an arrangement similar to that shown in 
Fig. 309, but plain washers are used in place of spherical seated 
ones. 

371. Collar Thrust Bearings. — (a) Marine thrust bearings. — 
For shafts subjected to a considerable end thrust, as for example 
a screw propeller shaft, or the shaft of a centrifugal pump or 
blower, the axial load is generally absorbed by a special type of 



544 



COLLAR THRUST BEARINGS 



[Chap. XIX 



thrust bearing, commonly called a collar thrust bearing. Instead 
of transmitting the axial load to the end of the bearing, the shaft 
is provided with a series of collars cut integral with it, which 
distribute the pressure over the length of the bearing. 

In modern marine practice the rings that come into contact 
with the collars on the shaft are made in the shape of a horseshoe. 
Such a construction permits their removal without disturbing 
any other part of the bearing. The lower part of the bearing 
housing is provided with a reservoir containing oil, and in order 
that the temperature of the oil may not become excessive, a water 
coil is fitted into this reservoir. Each end of the housing is 




Fig. 310. 



equipped with a stuffing box so that the level of the oi] in the 
reservoir may be carried slightly above the lower line of the shaft, 
thus insuring ample lubrication. Each of the bearing rings has 
an independently controlled circulation of water, thus making it 
possible to maintain a uniform temperature throughout the 
bearing. 

(b) De Laval thrust bearing. — For the high rotative speeds used 
on certain classes of centrifugal pumps, the De Laval Steam Tur- 
bine Co. developed a babbitt-lined ring-oiled collar thrust bear- 
ing, the details of which are clearly shown in Fig. 310. The 
collars, instead of being integral with the shaft, are formed on a 
removable steel or cast-iron sleeve which is fitted to the impeller 
shaft and held in place by a special collar and lock nut. The 



Art. 371] 



COLLAR THRUST BEARINGS 



545 



babbitt-lined bearing shells are split vertically, while the pedestal 
or housing into which these shells are fitted is split horizontally. 
From Fig. 310 it is evident that the cap of the bearing is not sub- 
jected to a thrust, but the entire axial load comes upon the 
pedestal. Vertically split shells such as those used on the De 
Laval bearing are easily removed, and since the two shells are 
alike they may be interchanged. The oil rings are made of 
bronze and a sufficient number are provided to insure ample 
lubrication. 

(c) Bearing for combined radial and axial loads. — In Fig. 311 
is shown a design of a combined ring-oiling journal and collar 




Fig. 311. 



thrust bearing that is used on a single suction multistage turbine 
pump. The bearing^ is babbitt-lined throughout and ample 
lubrication is furnished by means of an oil ring. The housing or 
bracket a, into which the combined bearing shells b and c are 
fitted, is cored out so that water may be circulated through it in 
order to keep the bearing cool. The tapped hole e at the top is 
connected to the discharge side of the first stage while the hole / 
at the bottom is connected to the pump suction. The shell of the 
bearing is split horizontally. To provide means for taking up 
the wear of the thrust collars, an adjusting screw g and lock nut 
is provided. This adjusting screw is also used for locating the 
propeller shaft h in its correct position relative to the guide vanes 
of the pump. 



546 



STEP BEARINGS 



[Chap. XIX. 



The thrust bearings discussed in the preceding paragraphs may- 
be located at any convenient point along the shaft, but precau- 
tions should be taken that the part of the shaft subjected to a 
thrust will not be too long or it may tend to fail by a buckling 
action similar to a long column. 

372. Step Bearings. — (a) Single-disc type. — Frequently a form 
of thrust bearing is used in which all the thrust must be taken up 
at the end of the shaft, as for example a vertical transmission 




(a) 




Fig. 312. 



shaft. For slow speeds such as prevail in rotary cranes of the 
jib and pillar types, the thrust due to the load and weights of 
moving parts are usually taken care of by an ordinary flat pivot 
or step bearing similar to the designs shown in Fig. 312 and 313. 
The thrust bearing illustrated in Fig. 312(a) is used on jib 
cranes and is frequently called a pintle bearing. The pintle or 
pin is subjected to a radial load in addition to the axial thrust. 
The pin in the design represented by Fig. 312(6) is also subjected 
to a combined radial and axial load and is used on pillar cranes. 



Art. 372] 



STEP BEARINGS 



547 



(6) Multiple-disc type. — The wear upon a pivot may be reduced 
materially by introducing several discs between the end of the 
pin or shaft and the housing of the bearing. Alternate discs are 
generally made of bronze or brass and steel. The lower disc 
should be fastened to the bearing proper, and the upper one 
should be fastened to the shaft, while the intermediate ones must 
be free. It is evident that each disc is subjected to the same unit 
pressure, hence the effect 
of such a combination of 
discs is to reduce the wear, 
since the relative velocity 
between adjacent discs is 
decreased. In order to 
lubricate the various disc 
surfaces, oil is introduced 
through a central hole and 
radial grooves cut into the 
faces of the discs serve as 
distributors. 

An application of the 
use of loose thrust discs is 
shown in Fig. 313 which 
illustrates a special step 
bearing designed by the 
Pawlings Harnischfeger Co. 
and used for supporting a 
heavy cantilever jib crane. 
In bearings of this kind 
the base casting is usually 
made in one piece, but in 

this case it is made in two parts, the base a and the cap b, 
which are bolted together by heavy stud bolts and special cap 
screws. The bronze bushed bearing shell c is provided with two 
spherical seats, the centers of which are located at the center of 
the bearing, as shown in the figure, thus insuring proper align- 
ment at all times. The thrust due to the load upon the crane 
and the weight of the crane comes upon the discs, two of bronze 
and one of steel, and is transmitted through the end of the shell 
c to the spherical seated pivot bearing in the base a. The hori- 
zontal pressure due to the load and weight is transmitted to the 
spherical journal bearing in the base a and cap b. The method 




Fig. 313. 



548 



FRICTION OF PIVOTS 



[Chap. XIX 



of lubricating the loose discs is clearly shown in the figure, also 
the method used for fastening the bushing e in the shell c. 

In Fig. 314 is shown a form of an adjustable step bearing that 
is intended for use at the base of a vertical shaft. It is evident 
from the construction that such a bearing cannot take care of 
heavy radial loads. Solid or split bearings must be provided 
for the radial loads. The bearing shell is babbitted and the 
axial load comes upon two hardened steel discs having spherical 
faces as shown in the figure. The housing containing the 
bearing is large and provides ample reservoir capacity for the 
lubricant. 



i 


i 


-J 


^ 


■ \\&k^ 


yj$ I ^ 




Fig. 314. 



373. Work Lost due to Pivot Friction. — General equations. — 
For the general case we shall assume the pivot to be some surface 
of revolution, as shown in Fig. 315(a), the equation of the curve 
being unknown. Assume any point Cat a distance x from the 
axis AB. If p denotes the intensity of the normal pressure at 
the point C, the total pressure on an annular strip of width ds 
and radius x will be 2 irxpds. Since the normal to the surface at 
the point C makes an angle 6 with the axis AB, the vertical 
component of the pressure oh the annular strip is 

dP = 2 wpxcosdds (610) 

If ri and r 2 denotes respectively the smaller and larger radii 
of the pivot, the integration of (610) between these limits will 
give the sum of the vertical components and this sum must be 
equal to the axial load or thrust P; that is 

P = 2x jpxdx (611) 

The value of the integral will depend upon the law of variation 
of the normal pressure p. 



Art. 373] 



FRICTION OF PIVOTS 



549 



The force of friction upon an annular strip of width ds is 
2-n-npxds, in which n denotes a coefficient of friction; hence the 
moment of this frictional resistance about the axis of rotation 

is 

dM = 2Trfxpx 2 ds, (612) 

from which we obtain the following general expression for the 
moment of friction of a pivot: 

M = 2Trffxpx 2 ds (613) 

From (613) it is evident that the value of M depends upon the 
following important considerations : 




i<*J 




Fig. 315. 



1. Upon the form of the pivot, that is upon the equation of the 
bounding curve. 

2. Upon the law of variation of the normal pressure p. 

3. Upon the law of variation of the coefficient of friction fx. 
In any given case the form of the pivot is known, but the laws 

of variation for p and y. are not known. But little experimental 
work has been carried on to establish such laws. A common 
method of dealing with pivots is to assume that the coefficient of 
friction remains constant and that the pressure is uniformly 
distributed. The assumption of uniform pressure distribution 
may represent fairly well the condition existing when the pivot 
and its bearing are new, but would seem unwarranted in the 
case of a pivot that has been worn in. A more reasonable sup- 



550 FRICTION OF COLLAR THRUST BEARINGS [Chap. XIX 

position is that the normal wear at any point is proportional to 
the work of friction. 

374. Work Lost in a Collar Thrust Bearing. — (a) Pressure 
uniformly distributed. — With the assumption that the normal 
pressure p is the same at all points of the surfaces in contact, the 
magnitude of the thrust P upon a collar pivot according (611) 
is given by the following expression: 

P=T P (rl-rl) ■ (614) 

From (614) it is apparent that the uniformly distributed pressure 
p is equal to the thrust P divided by the area of the collar. 

Substituting in (613) the value of p obtained from (614) and 
integrating, assuming n as constant, we obtain the following 
expression for the moment of friction of a collar bearing : 

3 L r 2 + n J 

The work, in foot-pounds, lost per minute by a collar pivot 
according to the above assumption may be determined by the 
formula 

_ gag H + ^ + ^ j 

9 L r 2 + n J 

in which the dimensions r 2 and r\ are expressed in inches and N 
denotes the revolution of the collar per minute. 

(b) Uniform vertical wear. — Letting n denote the normal wear 
of the collar, the statement "the normal wear at any point is 
proportional to the work of friction " L may be expressed by the 
relation 

n = kpx, 
from which 

' " 5 ^ 

Substituting this value of p in (611) and (613) and assuming n 
as constant, we obtain the following expression for P and M: 

P = 2j ?(r 2 - r.) (618) 



M = ^ {r\ - r\) (619) 



Art. 376] TOWER'S EXPERIMENTS 551 

Eliminating n and k between (618) and (619), we find that 

M = *f (r 2 + n), (620) 

which shows that the moment of friction of a collar pivot is the 
same as that of a ring of infinitesimal breadth, with a diameter 
equal to the mean diameter of the pivot. 

Upon the foregoing assumption, the work, in foot-pounds, lost 
per minute by a collar pivot is 

W f = ^ (r 2 + n) (621) 

375. Analysis of a Flat Pivot. — It is of interest to consider 
briefly the theoretical distribution of pressure in the case of a 
pivot in which the surface in contact is not a ring but a com- 
plete circle. From (617) we have 

n 

The normal wear n may practically be assumed as constant, hence 
the pressure p at any point varies inversely as the distance of 
that point from the axis of the pivot. Theoretically the pressure 
at the axis is infinitely great. While this is not the actual state 
of affairs, there is doubtless a great intensity of pressure at the 
axis and this produces a crushing of the material as experience 
with flat pivots seems to show. It is a good plan therefore to cut 
out the material at the center of the pivot as shown in Fig. 315(&) 
thus changing its surface of contact to that of a ring, as in the case 
of a collar pivot. The curve mn in Fig. 315(6) is an equilateral 

hyperbola whose equation is px = t, and it also shows graphic- 
ally how the pressure upon the contact surfaces varies. 

376. Tower's Experiments on Thrust Bearings. — (a) Collar 

bearings. — In the Proceedings of the Institution of Mechanical 
Engineers, 1888, p. 173, Mr. Beauchamp Tower reported the 
results of a series of experiments on a collar thrust bearing 14 
inches outside diameter and 12 inches inside diameter. The 
surfaces in contact consisted of a mild-steel ring located between 
two rings made of gun metal. Table 99, giving the values of the 
coefficient of friction for the various speeds listed, was compiled 
from the results published in the original report. The coefficients 



552 

Table 



TOWER'S EXPERIMENTS 



[Chap. XIX 



-Coefficients of Friction for Collar Thrust Bearings 
— Tower 



Pressures 


Revolutions per minute 




Total 


lb. 

sq. in. 


50 


70 


90 


110 


130 


values 


600 


14.7 


0.0450 


0.0646 


0.0433 


0.0537 


0.0642 


0.0541 


1,200 


29.4 


0.0375 


0.0481 


0.0496 


0.0489 


0.0475 


0.0463 


1,800 


44.1 


0.0357 


0.0399 


0.0361 


0.0357 


0.0371 


0.0369 


2,400 


58.8 


0.0286 


0.0375 


0.0361 


0.0373 


0.0410 


0.0361 


2,700 


66.1 


0.0354 


0.0334 


0.0346 


0.0361 


0.0378 


0.0354 


3,000 


73.5 


0.0347 


0.0341 


0.0348 


0.0352 


0.0356 


0.0348 


3,300 


80.8 


0.0337 


0.0322 


0.0348 


Bearing 


0.0336 


3,600 


88.2 


0.0312 


0.0444 




seized 


0.0378 



of collar friction as determined by Tower are based upon the 
assumption that the force of friction was concentrated at the end 
of the mean radius of the collars. From the results given in 
Table 99 it is apparent that the coefficient of friction is practi- 
cally independent of the speed and that it tends to decrease as 
the load on the bearing is increased. 

(6) Step bearings. — In Table 100 are given the results of a series 
of experiments, made by Tower, on a flat steel pivot 3 inches in 
diameter running on a manganese bronze step bearing. To in- 
sure proper lubrication of the contact surfaces, the oil was sup- 
plied to the center of the pivot and distributed by a single dia- 
metrical groove which extended to within }{ 6 inch from the 
circumference of the pivot. At the slower speeds the oil circula- 
tion, which was automatic due to the centrifugal action, was 



Table 100. — Coefficients 


of Friction for Step Bearings 


— Tower 






Revc 


lutions per minute 




lb. per sq. in. 


50 


128 


194 


290 


353 


20 


0.0196 


0.0080 


0.0102 


0.0178 


0.0167 


40 


0.0147 


0.0054 


0.0061 


0.0107 


0.0096 


60 


0.0167 


0.0053 


0.0051 


0.0078 


0.0073 


80 


0.0181 


0.0063 


0.0045 


0.0064 


0.0063 


100 


0.0219 


0.0077 


0.0044 


0.0056 


0.0057 


120 


0.0221 


0.0083 


0.0052 


0.0048 


0.0053 


140 




0.0093 


0.0062 


0.0046 


0.0053 


160 




0.0113 


0.0068 


0.0044 


0.0054 



Aet. 377] SCHIELE PIVOT 553 

somewhat restricted, varying from 20 to 56 drops per minute, 
while at the higher speed the bearing was flooded. Due to the 
more effective lubrication of the step bearing used in these experi- 
ments, the coefficients of friction are much less than those obtained 
with the experimental collar bearing. The coefficients of fric- 
tion as given in Table 100 were determined from the moments of 
friction by means of a formula based on the assumption that the 
pressure is uniformly distributed. In other words the moment 
of friction is M = % pPr. 

In a second series of experiments, a white metal step bearing 
was used in place of the manganese bronze bearing, and the results 
obtained gave coefficients of friction slightly greater than those 
given in Table 100. For all practical purposes the coefficients 
given in Table 100 may also be used for white metal bearings. 

377. Schiele Pivot. — It is possible to design a pivot with a sur- 
face of such a nature that the pressure between the pivot and its 
bearing shall be the same at all points of contact. From Fig. 
315(a), it is evident that the relation between the normal wear 
n and the vertical wear h is 

n = h cos0 (622) 

Combining (617) and (622), we obtain the following general ex- 
pression for the normal pressure : 

V = ^ (623) 

in which C denotes the ratio of the constants h and k. 

Assuming that p is to remain constant at all points of contact, 
it follows that cos is proportional to x, that is 

cos = Kx (624) 

Since is the angle that the normal to the bounding curve of the 
pivot makes with the axis of the pivot, we have 

dy = tanddx 
Differentiating (624) 

• Kdx = — sinddd, 
whence 



Integrating (625), we get 

Ky = sin - log, (sec + tan 0) + F 



554 SCHIELE PIVOT * [Chap. XIX 

Eliminating 6 by means of (624) 

"l ± VT~^kw 



Ky = Vl - KH 2 - log, 



Kx 



? ] 



+ F (626) 



This is the equation of the tractrix or sometimes wrongly called, 
the antifriction curve. 

From Fig. 315(a), cos0 = -^, and combining this with (624), 

we find K = ^^. From the construction of the tractrix BC = r 2 , 
therefore 

K = i (627) 

Moment of friction. — The moment of friction, about the axis 
of the pivot, of the normal pressure p acting on an annular strip 
of width ds is 

dM = 2nirpr2xdx 

Assuming as in the preceding discussions that the coefficient n 
remains constant, we obtain by integration the following ex- 
pression for the moment of friction : 

M = M 7rpr 2 (rl - rl) (628) 

For uniform distribution of pressure it was shown that 

P 

P / 2 2\ 

ir (4 - rl) 

Substituting this value of p in (628), we have 

M = juPr 2 (629) 

Comparing (629) with the expression for the moment of fric- 
tion for a collar pivot as given by (620), it is evident that the 
moment of friction of a Schiele pivot is always the greater. The 
Schiele pivot has one advantage in that it keeps its shape as it 
wears and it is self-adjusting. Due to its excessive cost of manu- 
facture it is used but little. 

References 

Elements of Machine Design, by W. C. Unwin. 

Bearings and Their Lubrication, by L. P. Alford. 

Friction and Lost Work in Machinery and Mill Work, by R. H. Thurston. 

Lubrication and Lubricants, by L. Archbutt and L. M. Deeley. 



Art. 377] REFERENCES 555 

Handbook for Machine Designers and Draftsmen, by F. A. Halsbt. 

Theory of Lubrication, Phil. Trans., 1886, Part I, p. 157. 

Report on Experiments on Journal Friction, Proc. Inst, of Mech. Engrs., 
1883 and 1885. 

Report on Experiments on Collar Friction, Proc. Inst, of Mech. Engrs., 
1888, p. 173 

Report on Experiments on Pivot Friction, Proc. Inst, of Mech. Engrs., 
1S91, p. 111. 

Bearings, Trans. A. S. M. E., vol. 27, p. 420 

Comparative Test of Three Types of Lineshaft Bearings, Trans. A. S. M. 
E., vol. 35, p. 593. 

On the Laws of Lubrication of Journal Bearings, Trans., A. S. M. E., vol. 
37, p. 167. 

Friction Tests of Lubricating Greases and Oils, Bull. No. 4, Univ. of Mo., 
December, 1913. 

Bearings for High Speed, Traction and Transmission, vol. 6, No. 22. 

Experiments, Formulas and Constants of Lubrication of Bearings, Amer. 
Mach., vol. 26, pp. 1281, 1316, 1350. 

Charts for Journal Bearings, Amer. Mach., vol. 37, p. 848. 

Charts for Journal Bearings, Amer. Mach., vol. 39, pp. 1017 and 1069. 

Lubrication of Bearings, Amer. Mach., vol. 45, p. 847. 

Temperature Tests on Journal Bearings, Power, vol. 37, p. 848. 

Bearing Design Constants, Power, vol. 43, p. 251. 

Electrical Machine Bearings, Power, vol. 44, p. 340 

Pressure Oil-Film Lubrication, Power, vol. 44, p. 798. 

Experiments with an Air-lubricated Journal, Jour. A. S. Nav. Engrs., vol. 
9, No. 2. 

The Kingsbury Thrust Bearing, The Electric Journal. 

A New Type of Thrust Bearing, Trans. Nat. Elect. Lt. Assoc, 1913. 



CHAPTER XX 
BEARINGS WITH ROLLING CONTACT 

378. Requirements of Rolling Contact. — A bearing having a 
rolling contact is one in which the journal is supported on rollers 
or balls, thereby decreasing the frictional resistance, since rolling 
friction seldom exceeds sliding friction under the same conditions 
of load and operation. Due to the use of improved machinery 
for producing the rolling elements, bearings with rolling contact 
are now used for all classes of service. A bearing of this kind to 
be commercially successful must fulfill the following conditions : 

(a) The arrangement of the rolling elements should be such 
that sliding is reduced to a minimum. 

(6) The rolling elements must all be of the same size, and ac- 
curacy in form is absolutely essential. 

(c) The rolling elements must be extremely hard and their 
surfaces must be polished very smooth. 

(d) The rolling elements must be so arranged that they will 
not run off their guides or raceways. 

(e) The rolling elements must not be overloaded, as they may 
become distorted thus changing the conditions entirely. 

(/) The pressure should be approximately normal to the sur- 
face of contact. 

379. Classification. — Bearings with rolling contact may be 
divided, according to the kind of rolling element used, into the 
following classes: 

(a) Roller bearings, in which either cylindrical or conical rollers 
are placed between the journal and its bearing. 

(6) Ball bearings, in which hardened steel balls are used in 
place of the rollers. 

Each of the above classes may be subdivided into the following 
types: (1) radial bearings; (2) thrust bearings. 

ROLLER BEARINGS 

380. Radial Bearing having Cylindrical Rollers. — (a) Moss- 
berg bearing. — The simplest form of roller bearing for a journal 

556 



Art. 380] 



RADIAL ROLLER BEARINGS 



557 



consists of a sleeve surrounded by a series of cylindrical rollers, 
rolling inside of a bored casing or outer race. Fig. 316, shows such 
a bearing made by the Standard Machinery Co. and known as 
the Mossberg bearing. The sleeves, rollers, and outer casings 
must have true cylindrical surfaces and for proper working the 
axes must always remain parallel to each other. To keep the 
rollers d in the desired position, they are placed in a cage c 
having its outside diameter slightly less than the inner diameter 
of the outer race e, and its internal diameter a trifle greater than 
the diameter of the sleeve b. The cage, made from a good tough 
bronze or steel, is provided with a series of slots reamed to size, 





Fig. 316. 



into which the steel rollers are placed as shown in Fig. 316. 
No doubt there is a certain amount of sliding between the roller 
and the cage, but actual tests seem to show that this sliding 
action reduces the efficiency of the bearing but little. 

(b) Norma bearing. — A form of roller bearing shown in Fig. 
317 has been recently developed and placed on the market. It 
consists of an outer race e having a convex or ball-shaped interior 
surface, against which the rollers d bear. The sleeve or race b 
is cylindrical and is fastened to the shaft or journal. The 
cylindrical rollers, which are short, are held in alignment by the 
specially constructed steel cage c. As may be seen in Fig. 317, 
the outer race e is open-sided thus facilitating the assembling, 
mounting or dismantling of the bearing. The manufacturers of 
this bearing, which originated in Stuttgart, Germany, claim that 



558 



RADIAL ROLLER BEARINGS 



[Chap. XX 




the Norma bearing, as it is called, is capable of supporting greater 
loads than ball bearings having the same dimensions and running 
at the same speed. These bearings are made so that they are 
interchangeable with ball bearings, thus providing for their 
application in cases where ball bearings fail 
under the applied load. 

Due to inaccuracies of the rolling element or 
to wear, the rollers in an ordinary roller bear- 
ing may acquire a tendency to move length- 
wise, thus causing more or less end pressure 
on the cage. To eliminate this end pressure, 
holes are drilled in the ends of the rollers, or in 
the cage, and steel balls are inserted. 

381. Radial Bearings having Conical Rollers. 

— The rollers instead of being cylindrical may 
be conical as shown in the bearing illustrated in 
Fig. 318. In its general construction this bear- 
ing is similar to the plain roller bearing. It 
consists of a series of conical rollers d located between the inner 
and outer cones b and e. The cage, consisting of two rings c and / 
made of high-carbon steel, is provided with sockets for holding 
the ends of the rollers. These rings are held together by stay rods 
g, shown by the dotted lines. The ends of the rollers are beveled 






Fig. 319. 



to aVigh t angle and bear against corresponding shoulders on the 
cage*and inner cone 6. To insure true rolling in this type of 
bearing it is necessary that all the axes of the rollers intersect 
the axis of the journal in a common point. Bearings having 
two'sets of conical rollers are also made at the present time. 



Art.. 382] 



HYATT BEARING 



559 



Timken roller bearing. — Another successful roller bearing using 
tapered rollers is shown in Fig. 319. It is used rather extensively 
in automobile construction and differs from the bearing just 
described in minor details only. The Timken bearing is made in 
various styles and that shown in Fig. 319 is known as the "short 
series." 

One important advantage possessed by conical roller bearings 
over any other form of roller bearing lies in the provision for 
taking up the wear if there is any. It is merely necessary to 
force the inner cone and the rollers further into the outer cone 
or cup. 

382. Radial Bearings having Flexible Rollers. — Due to in- 
accuracy in manufacturing, slight deflections of the shaft, yield- 





Fig. 320. 



ing of the supports or mounting of a roller bearing, a roller may 
move out of its correct position and cause the line of contact 
with the sleeves or races to become curved instead of straight. 
Such a condition would cause a long roller of brittle material to 
break and the whole bearing would thereby be ruined. To over- 
come this difficulty the flexible roller has been devised and is 
now used for all classes of service. A form of bearing, known as 
the Hyatt bearing, using flexible rollers is shown in Fig. 320. 
The Hyatt rollers are made of a strip of steel wound into a coil 
or spring of uniform diameter. Due to its flexibility, the roller 
will adjust itself to any irregularity of the bearing such as im- 
perfect alignment; furthermore, the distribution of the load along 



560 



ROLLER THRUST BEARING 



[Chap. XX 



the entire length of the roller will be practically uniform, thus 
permitting the use of commercial shafting, hardened and ground 
journals not being essential except under extreme conditions of 
loading. Due to its construction the lubrication of the Hyatt 
bearing is very effective, since the center of each roller is really 
a large oil reservoir. The Hyatt bearing is also made with a 
hardened steel inner sleeve which may be fastened to a soft 
steel shaft. 

383. Thrust Bearings having Cylindrical Rollers. — In Fig. 
321 is shown a thrust bearing using cylindrical rollers. It is 




Fig. 321. 



claimed by the manufacturer of this bearing, that while theoretic- 
ally a thrust bearing having conical rollers is better than one 
having cylindrical rollers, the theory is not borne out in actual 
operation. The explanation no doubt lies in the mechanical in- 
accuracy of the various parts in contact. To reduce the tendency 
of the cylindrical rollers to groove the discs, the rollers should 
travel in different paths. In some designs this is accomplished by 
placing the slots in the cage at different distances from the shaft 
center. Another scheme used for preventing the formation of 
grooves is shown in Fig. 321, and consists of rollers having dif- 
ferent widths. The thrust of the rollers in a radial direction 
may be taken up by a ball, as shown in the figure. 



Art. 384] 



ROLLER THRUST BEARING 



561 



Fig. 321 shows a roller thrust bearing installed under a 5,500- 
horse-power turbine generator of the Niagara Falls Power Co. 
The maximum load coming upon this bearing is 190,000 pounds 
and the normal load is 156,000 pounds. The normal speed is 
250 revolutions per minute, and the maximum may reach 500 
if the governor fails. By consulting Fig. 321 it will be noticed 
that on the under side of each cage c and near its bore is located 
an auxiliary roller bearing, the function of which is to support 
the weight of the cage. The cages of both the main and auxiliary 
bearings are made of bronze, while the thrust discs or washers are 
made of case-hardened machinery steel. 

384. Thrust Bearings having Conical Rollers. — A common 
form of thrust bearing using conical rollers consists of a cage c, a 
series of steel rollers d, two steel thrust discs e and b, and an exter- 
nal ring / as shown in Fig. 322. 

The cage c, generally made of 
one solid piece of metal, is pro- 
vided with tapered holes into 
which are placed the conical 
rollers d. Due to the action 
of the load upon the bearing, 
the rollers tend to move radi- 
ally outward, and to reduce 
this tendency, the apex angle 

is made relatively small. One prominent manufacturer makes 
this angle 6 degrees. The thrust discs may both be made coni- 
cal, or either may be flat and the other coned; in other words, 
the axes of the rollers need not necessarily be at right angles to 
the axis of the shaft. However, to obtain pure rolling, the ver- 
tices of the rollers and of the conical thrust surfaces must be in 
a common point on the axis of the shaft. In the thrust bear- 
ing shown in Fig. 322, the radial thrust of the rollers is taken 
care of by the tool-steel ring /. In some designs the end thrust 
of the rollers against the cage is taken by a ball located between 
two cupped surfaces. 

385. Allowable Bearing Pressures and Coefficients of Friction. 

— The intensity of pressure coming upon the elements of a roller 
bearing should not exceed the elastic limit of the material or 
permanent deformation will occur. Such deformation ruins 
either the rollers or the bearing surfaces, or both. 




562 ROLLER BEARING DATA [Chap. XX 

In 1898 Prof. Stribeck, the well-known head of the Technical 
Laboratories in New Babelsberg carried on extensive investiga- 
tions on sliding, roller and ball bearings. Among the tests made 
was a series investigating the relations existing between the 
coefficient of friction, specific load, and the speed for many types 
and sizes of bearings. The following are a few of the conclusions 
mentioned in a report submitted by Prof. Stribeck. 

(a) That the load coming upon either a roller or ball bearing 
may be considered as supported by one-fifth the number of rollers 
or balls in the bearing. This distribution of the load is not uni- 
form over each of the carrying rollers or balls. 

(6) That the ball bearing has a load carrying capacity much 
in excess of that of the roller bearing. 

(c) That the roller bearing has a higher coefficient of friction 
than the ball bearing for similar conditions of speed and loading. 

(d) That the coefficient of friction for ball bearings is practic- 
ally a constant for a wide range of speed and load. 

(e) That the chief advantage of roller bearings over plain bear- 
ings lies in the lower coefficient of friction. 

By the term "specific load" is meant the pressure per unit of 
carrying element. For a plain bearing the carrying element is 
considered the projected area of the journal. For roller bearings 
the carrying element is considered equivalent to one-fifth of the 
number of rollers times the product of the length by the dia- 
meter of the rollers. For ball bearings, the product of one-fifth 
of the number of balls and the square of the diameter is considered 
as equivalent to the carrying element. 

386. Roller Bearing Data. — Roller bearings for motor car ser- 
vice have been standardized to such an extent by several manu- 
facturers that they may be interchanged with ball bearings of 
similar capacity. 

(a) Norma bearings. — In Table 101 are given the various di- 
mensions of the medium and heavy-duty Norma roller bearings. 
The symbols denoting the dimensions refer to the key drawing of 
Fig. 317. The load capacity as given in this table is based on a 
steady load and slow speed. To obtain the rating at any partic- 
ular speed, multiply the rating given in Table 101 by the speed 
coefficient obtained from Fig. 323. The chart plotted in this 
figure is based upon data deduced from the load ratings given in 
the trade publication issued by The Norma Co. of America. In 
addition to the types of Norma bearings listed in the table, a 



Art. 386] ROLLER BEARING DATA 563 

Table 101. — Data Pertaining to Norma Roller Bearings 



Medium-duty series 






Ilea 


vy-dutj 


scries 






Dimensions 


Load 






Dimensions 


Load 




e 




at 
10 


Size 








at 




















10 






1 


2 


3 


4 


r.p.m. 






1 


2 


3 


4 


r.p.m. 


NM 


25 


2.4410 


0.9842 


0.67 


0.04 


1,650 


NS 


25 


3.1496 


a> 


0.830.08 


3,410 


NM 


30 


2.8346 


1.1811 


0.75 


0.08 


2,150 


NS 


30 


3.5433 




0.91 0.08 


3,850 


NM 


35 


3.1496 


1.3779 


0.83 


0.08 


2,750 


NS 


35 


3.9370 


3 


0.98 0. OS 


4,400 


NM 


40 


3.5433 


1.5748 


0.91 


0.08 


3,520 


NS 


40 


4.3307 


S 


1.06 


0.08 


5,940 


NM 


45 


3.9370 


1.7716 


0.98 


0.08 


4,400 


NS 


45 


4.7244 


1.14 


0.08 


7,260 


NM 


50 


4.3307 


1.9685 


1.06 


0.08 


5,280 


NS 


50 


5.1181 


TJ 


1.22 


0.08 


8,580 


NM 


55 


4.7244 


2.1653 


1.14 


0.08 


6,600 


NS 


55 


5.5118 


a 


1.30 


0.12 


9,460 


NM 


60 


5.1181 


2.3622 


1.22 


0.08 


7,700 


NS 


60 


5.9055 


a 


1.38 


0.12 


11,200 


NM 


65 


5.5118 


2.5590 


1.30 


0.12 


8,360 


NS 


65 


6.2992 


IN 


1.45 


0.12 


12,100 


NM 


70 


5.9055 


2.7559 


1.38 


0.12 


10,120 


NS 


70 


7.0866 


a 


1.65 


0.12 


16,060 


NM 


75 


6.2992 


2.9527 


1.46 


0.12 


11,880 


NS 


75 


7.4803 


"i 


1.77 


0.12 


18,040 


NM 


80 


6.6929 


3.1496 


1.54 


0.12 


12,980 


NS 


80 


7.8740 


J 


1.88 


0.12 


18,260 


NM 


85 


7.0866 


3.3464 


1.61 


0.12 


14,080 


NS 


85 


8.4645 


2.00 


0.12 


19,140 


NM 


90 


7.4803 


3.5433 


1.69 


0.12 


16,280 


NS 


90 


8.8582 




2.12 


0.12 


23,320 


NM 


95 


7.8740 


3.7401 


1.77 


0.12 


17,380 


NS 


95 


9.6456 


4) 


2.24 


0.12 


27,280 


NM 100 


8.4645 3.9370 


1.85 


0.12 


21,120 


NS 100 


10.4330 


02 


2.36 


0.12 


37,400 



1.0- 

.£ 0.9 

c 

<V 
CO 

k 0.8 

s. 

1 

£ 0.7 

z 

t_ 

s 


































o 
o 

"8 0.5 

QJ 
Q. 

0.4 







































































































per mm. 
Fig. 323. 



light-duty bearing is also manufactured; furthermore, each of the 
three types is also made in larger sizes than those given. 



564 



MOUNTING OF ROLLER BEARINGS 



[Chap. XX 



(&) Hyatt bearings. — The dimensions and load-carying capaci- 
ties for the long and short series of the Hyatt high-duty type of 
roller bearing, similar to that shown in Fig. 320, are given in 
Table 102. The type of bearing to which the data given in this 

Table 102. — Data Pertaining to Hyatt High-duty Bearings 



Short series 




Long 


series 








Dimensions 


Rating 


Size 


Di 


mensions 




Size 














Rating 




1 


2 


3 






1 


2 


3 




17,010 


1.000 


2.249 


1 .000 


460 


17,060 


2.000 


.2 


02 
.2 


1,200 


17,012 


1.000 


2.374 


1.125 


500 


17,062 


2.000 




03 

02 


1,340 


17,014 


1.125 


2.749 


1.250 


700 


17,064 


2.250 


-£ 


t, 


1,700 


17,016 


1.125 


2.875 


1.375 


750 


17,066 


2.250 


o 


O 


1,900 


17,018 


1.250 


3.375 


1.500 


960 


17,068 


2.500 


QJ 




2,340 


17,020 


1.250 


3.499 


1.625 


1,040 


17,070 


2.500 


■5 


2,530 


17,022 


1.250 


3.625 


1.750 


1,125 


17,072 


2.500 


o 


>S 


2,730 


17,024 


1.250 


3.749 


1.875 


1,200 


17,074 


2.500 




CO 


2,925 


17,026 


1.375 


4.124 


2.000 


1,470 


17,076 


2.750 


3,490 


17,028 


1.375 


4.249 


2.125 


1,550 


17,078 


2.750 


a 


a 


3,700 


17,030 


1.375 


4.374 


2.250 


1,650 


17,080 


2.750 


I 


i 


3,925 


17,032 


1.500 


4.749 


2.500 


2,060 


17,082 


3.000 


-3 


T3 


4,820 


17,034 


1.500 


4.999 


2.750 


2,270 


17,084 


3.000 


3 


03 


5,300 


17,036 


1.750 


5.374 


3.000 


3,030 


17,086 


3.500 


S 


0) 

a 


6,890 


17,038 


1.750 


5.624 


3.250 


3,400 


17,088 


3.500 


03 


OS 


7,600 



table applies necessitates the use of a heat-treated or hardened 
steel shaft, since no inner shell or sleeve is furnished. The load 
ratings specified in Table 102 represent the load in pounds that 
any particular bearing is capable of carrying at a speed not to 
exceed 1,000 revolutions per minute. According to the manu- 
facturer of the Hyatt bearings, the load capacity at 1,500 revolu- 
tions per minute should be taken as equivalent to 50 per cent, 
of that given in the table, and when the speed is 500 revolutions 
per minute the capacity may be increased 50 per cent, above that 
given for 1,000 revolutions per minute. 

387. Mounting of Roller Bearings. — To obtain satisfactory 
service, roller bearings of all types must be carefully protected 
from water, acids, alkalies, dust, and any foreign matter that 
might ruin them. Protection may be obtained by housing in the 
bearing and sealing the openings through which the shaft passes 
with felt packed into grooves provided for that purpose in the 



Art. 387] 



MOUNTING OF ROLLER BEARINGS 



565 



end or cover plates. Filling the housing and bearing with a 
high-grade stiff grease, provided the speed is not too high, will 
also aid in keeping out foreign matter, and at the same time it 
will furnish the necessary lubrication. 

In roller bearings of the Norma and Hyatt type both the inner 
and outer races or sleeves are rigidly held in place. Generally 
the inner race is made a light driving fit on the shaft, and to in- 
sure a rigid fastening, the race should be clamped between a suit- 
able shoulder on the shaft and a nut provided with some form of 
locking device. For various forms of nut locks consult Art. 80. 
The outer race is usually clamped between a shoulder in the hous- 
ing and an outside cover plate, or in some cases between two 
cover plates. The shoulders on the shafts or in the housing against 




Fig. 324. 



which a bearing is clamped should be sufficiently high to provide 
ample support to the bearing. If, for example, the shoulder on 
the shaft be made too small, the inner race when pressed into 
place is liable to slip over this shoulder and cause the race to 
expand slightly thus producing undue pressure upon the end of 
the roller. Since Norma and Hyatt bearings cannot take an end 
thrust, the latter must be taken care of by suitable thrust bearings. 
In mounting a conical roller bearing either the cone or the cup 
must be provided with means for taking up wear. When the 
inner race or cup is mounted on a non-rotating member, as for 
example on the front wheel spindle of a motor car, it is considered 
good practice to fasten the outer race or cup rigidly into the hub 



566 FORMS OF RACEWAYS [Chap. XX 

casting or forging, and provide the cone with an adjustment for 
taking up wear by making it an easy sliding fit on the spindle. 
When the cone is mounted on a rotating member, good practice 
dictates that the cone be made a tight press fit on the shaft and 
that the wear be taken up by making the cup adjustable. A good 
example of an installation of conical roller bearings, in which the 
cones are mounted on rotating members, is shown in the rear- 
axle worm-gear transmission of Fig. 324. Attention is directed 
to the fact that due to the rigid mounting of the outer races 
against the rim of the end cover plates, the worm shaft is always 
subjected to a compression, and any expansion of the shaft due 
to an excessive rise in temperature will cause it to deflect a small 
amount and necessarily produce undue wear. To obviate such 
a condition, the bearings may be so arranged that the cups will 
come against shoulders on the housing or gear case, thus causing 
the worm shaft to be in tension. 

BALL BEARINGS 

Formerly ball bearings were used chiefly for light loads, but 
at the present time they are used in all classes of machinery. In 
general, a ball bearing consists of a series of balls held by a suitable 
cage between properly formed hardened steel rings called races. 
These races may be of such shape that the ball has two points 
of contact, as shown in Fig. 325, or it may have three or even 
four points of contact, as shown in Figs. 326 and 327, respectively. 

388. Forms of Raceways. — (a) Two-point contact. — The sim- 
plest form of two-point contact is the flat race shown in Fig. 
325(a). In this construction no provision is made for retaining 
the balls. To overcome this objection, the races may be curved 
as shown in Fig. 325(6), (c) and (d), the latter having the greatest 
carrying capacity. This increase of carrying capacity is no 
doubt due to the increased area of contact. For a well-designed 
ball bearing the wear upon the inner and outer races should be 
the same, which means that the contact pressure upon these 
races should be the same. The contact pressure depends upon 
the small contact area, and if these areas are to be equal it is 
necessary that the radius of curvature of the outer race should 
be increased. This has been done in the bearing shown in Fig. 
325(d). 



Art. 



FORMS OF RACEWAYS 



567 



(b) Three-point contact. — In Figs. 326 and 327 are shown forms 
of bearing raceways having three and four points of contacts, 
respectively. To produce true rolling of the ball the races must 




(a) 



(b) 



Fig. 326. 



be laid out correctly. Referring to Fig. 326(6), and letting A, 
B and C represent the three points of contact, extend the line A B 
until it intersects the center of the shaft at 0, also draw OF 
through the center of the ball. This latter line represents the 



568 



CONCLUSIONS OF ST RI BECK 



[Chap. XX 



axis of rotation of the ball, and the lines AE and BD are projec- 
tions of two circles of rotation. From similar triangles we have 
that AE and BD are proportional to OA and OB respectively; 
therefore it is evident that there is no slipping at the points A 
and B and that the desired true rolling is obtained. . The third 
point of contact C is determined by drawing OC tangent to the 
ball. To avoid excessive wedging of the ball, the angle a should 
be made not less than 30 degrees. 

(c) Four-point contact. — -In either of the four-point bearings 
shown in Fig. 327, the pure rolling of the ball is obtained when 




(a) 




Fig. 327. 



^np — jyn The various lines required for laying out a bearing 

of this kind are drawn in a general way, according to the method 
outlined for the three-point bearing in the preceding paragraph. 

389. Experimental Conclusions of Stribeck. — Prof. Stribeck 
in his investigation of bearings having rolling contact determined 
how the carrying capacity of a ball was affected by the form of the 
raceway. For this purpose ball bearings having raceways shown 
in Figs. 325 to 327 inclusive, except the form shown in Fig. 325(d), 
were used. 

Some of the conclusions arrived at were as follows : 

(a) The form of raceway shown in Fig. 325(a) had the least 
frictional resistance. 

(6) An increase in the number of points of contact, as shown in 
Figs. 326 and 327, resulted in higher frictional resistances. It is 



Art. 390] RADIAL BALL BEARINGS 569 

probable that due to imperfect workmanship the conditions re- 
quired for pure rolling were not met. 

(c) The carrying capacities for the forms of raceways shown 
in Figs. 326 and 327 were practically the same. Theoretically 
the four-point contact should carry more, but due to difficulties 
in constructing and adjusting such a bearing it is almost impossi- 
ble to distribute the load uniformly over the various points of 
contact. 

(d) The carrying capacity for the form of raceways shown in 
Fig. 325(c) is considerably greater than for the other forms shown. 

(e) The frictional resistance for the form indicated by Fig. 
325(c) is a trifle greater than for the others, but practically it 
may be considered the same. 

390. Radial Ball Bearings. — (a) Single-row bearings. — A radial 
ball bearing is used for supporting loads acting at right angles to 
the axis of rotation. At the present time the two-point contact 
type having circular raceways is used almost exclusively. Such 
a bearing consists of an outer and inner race, both provided with 
curved ball raceways that are uniform and unbroken around the 
entire circumference. Between these races is located a series of 
balls separated either by an elastic separator or by a bronze or 
alloy cage, as shown in Fig. 325. The type of elastic separator 
mentioned consists of a short helical spring fitted with suitable 
bearing plates. This separator was formerly used in the Hess- 
Bright bearings and at the present time is still used under certain 
special conditions. The majority of the separators or cages now 
in use are made of brass or bronze and steel and their construc- 
tion makes them more or less elastic. 

(6) S. K. F. bearing. — Radial bearings having two or more 
rows of balls, examples of which are shown in Figs. 325(d) and 
328, have also been devised. In selecting this type of bearing 
it must not be assumed that doubling the number of balls neces- 
sarily doubles the load capacity, for the accuracy of workmanship 
required for such a condition is not always feasible. 

The bearing shown in Fig. 325(d) originated in Sweden and is 
known as the S. K. F. bearing. The outer ball race e is a ma- 
chined and ground spherical surface, the center of which lies on 
the axis of the bearing. The inner race b has two curved ball 
raceways having a radius slightly larger than the radius of the 
ball. The balls are staggered and are retained by the phosphor 
bronze separator or cage c. This type of bearing may be dis- 



570 



THRUST BALL BEARINGS 



[Chap. XX 



mantled very readily by swinging the race b and the balls together 
within the outer race e, and then removing two adjacent balls on 
either side diametrically opposite to each other. This operation 
permits the withdrawal of the complete center portion of the bear- 
ing. Another important advantage of the S. K. F. bearing is its 
self-aligning feature, which compensates for shaft flexure or 
deformation. 

(c) Norma bearing. — In Fig. 328 is shown a form of double-row 
ball bearing manufactured by The Norma Co. of America. It 

consists of two outer races 
• mounted side by side on a 

single inner race provided with 
two raceways. The raceways 
in the outer rings are ground 
to the same radius as that used 
on the inner race, but one-half 
of the shoulder is ground 
away to form a cylindrical 
surface, tangent to the circu- 
lar raceway, as shown in the 
figure. It is evident that this 
form of outer race differs 
materially from that shown in 
Fig. 325(c). The main ad- 
vantage of the construction 
used on the Norma bearing 
lies in the ease with which 
that bearing can be assembled 
and dismantled for inspection. 
The separator used consists 
of alight one-piece bronze ring 
having a channel section. The flanges of this ring separator are 
provided with spherical seats between which the balls are held 
with a slight elastic pressure; thus the balls and separator may 
be removed as a single unit. 

391. Thrust Ball Bearings. — (a) Two-point type. — The modern 
ball thrust bearing is made"with either two- or four-point contact. 
In Fig. 325(a) is shown a two-point contact having flat raceways. 
It consists of two hardened steel plates or thrust discs b and c be- 
tween which is located the cage c containing the balls. The 
cage may be made of one piece by drilling the holes for the balls 





^P 


'WfM 


1 












wm 






L 


.. 






_« ; 


5 — 





Fig. 328. 



Art. 391] 



THRUST BALL BEARINGS 



571 



almost through, then inserting the ball and by means of a special 
setting tool closing in the upper edge. The cage may also be 
made in two pieces as shown in the figure. 




Fig. 329. 

The type of thrust bearing having curved or grooved raceways 
is shown in Figs. 329 and 330(a). The constructive features are 
clearly shown in the figures. These bearings are known as the 




(*) 



(b) 



Fig. 330. 



full ball or without separator type, and are intended for very 
heavy service at a slow speed. The type shown in Fig. 330(a) 
being made in small sizes is intended for use on automobile steer- 
ing pivots, while that illustrated in Fig. 329 is made in the larger 



572 



THRUST BALL BEARINGS 



[Chap. XX 



sizes and is used on crane hooks. Another type of thrust bearing 
having curved raceways is shown in Fig. 330(6). The balls are 
separated by a cage made of brass or special alloy. 

All thrust bearings thus far shown are intended to take the 
thrust in one direction only. In cases where the thrust has to 
be taken care of in both directions, a form known as the double- 
thrust bearing is used. Such a bearing, shown in Fig. 331, 
consists of a central grooved disc / securely fastened to the shaft, 
two thrust discs b and e having grooves to correspond with those 
on /, and two phosphor-bronze cages c retaining the balls. The 




Fig. 331. 



form of bearing just described may be so arranged that the com- 
bination of balls, cages, and thrust disGS form a part of a sphere 
as shown in Fig. 331. The entire combination is then free to 
revolve in a specially constructed hardened steel casing g. To 
permit easy assembling two recesses are located in a convenient 
position on one side of the casing g. 

(b) Four-point type. — A four-point bearing made by the Auburn 
Ball Bearing Co. is shown in Fig. 332. All thrust bearings made 
by this company are of the four-point type and have no separator 
for the balls. The condition for pure rolling is fulfilled as may be 
seen from the geometry of the figure. 

(c) Leveling washer. — In any thrust bearing it is always desir- 
able to distribute the load uniformly over the entire series of 



Art. 392] 



RADIO-THRUST BEARINGS 



573 



balls. This is done by providing one of the thrust discs with a 
spherical surface thus permitting it to adjust itself. The con- 
struction is shown in Figs. 330 and 331. When both discs are 
flat as is the case of Figs. 329 and 332. a special leveling washer 
having a spherical seat should be used in connection with the 
stationary thrust disc. 

392. Combined Radial and Thrust Bearing. — A combination 
radial and thrust bearing is used in places where provision must 
be made for both radial and axial loads. Some of these bearings 
are so arranged that, in addition to the radial loads, they will take 
care of a thrust in only one direction or in both directions. 




Fig. 332. 



(a) Radax bearing. — A bearing known as Radax, manufactured 
by the New Departure Mfg. Co., is used for both radial and one 
direction axial loads. The details of this bearing are clearly 
shown in Fig. 325(6) . The bearing differs from an ordinary radial 
bearing in that the ball raceway has a two-point angular contact 
instead of radial contact. The Radax bearing may readily be 
assembled and dismantled since the inner race, separator, and 
balls may easily be withdrawn from the outer race. These bear- 
ings are made interchangeable with corresponding sizes of stand- 
ard radial bearings. 

(6) Gurney radio-thrust bearing. — In Fig. 333 are shown the 
details of a combined radial and thrust bearing manufactured by 
the Gurney Ball Bearing Co. The points of contact between 
the balls and the inner and outer races are not on radial lines, 
but lie on the lines that intersect the axis of the bearing at the 
point 0, as shown in the figure. The steel separator is made in a 
single piece having a light but rigid construction. The radio- 
thrust bearing is well adapted to installations in which there is 
a combination of radial and axial loads, and where the latter ex- 



574 



RADIO-THRUST BEARINGS 



[Chap. XX 



ceeds approximately 25 per cent, of t^e former. It is generally 
conceded that ordinary radial bearings should- not be subjected 
to an axial load exceeding 25 per cent, of the radial load. With 
relation to thrust capacity, the Gurney radio-thrust bearings 
are made in three types. Each of these types is made in three 
series, namely, the light, medium, and heavy, and as far as the 
dimensions are concerned these bearings are interchangeable 
with the corresponding sizes of standard radial bearings. 

(c) Double-row and duplex bearings. — The bearings discussed 
in the preceding paragraphs are used in places where the axial 





Fig. 333. 



Fig. 334. 



loads are always in the same direction. There are, however, 
many places requiring bearings capable of taking a thrust in 
either direction. Such a condition can be successfully met by 
installing Duplex bearings which consist of two radio-thrust bear- 
ings mounted side by side, or by using a double-row combined 
radial thrust bearing. A bearing of the latter type, made by the 
New Departure Mfg. Co., is shown in Fig. 334. It consists of a 
single inner race containing two raceways, a bronze separator 
made in two pieces, two rows of balls having two-point angular 
contact, two outer races, and a thin steel shell which is closed in 
over the outer races, as shown in the figure, after the bearing is 
assembled. 



Art. 393] 



BEARING PRESSURES 



575 



393. Allowable Bearing Pressures. — (a) Load per ball. — Ac- 
cording to Stribeck, the carrying capacity in pounds per ball may 
be determined by the formula 



= kd* 



(630) 



in which d denotes the diameter of the ball in inches, and k 
a constant depending upon the form and material of the raceway 
and the speed of the bearing in which the ball is used. The 
following values of k, due to Stribeck, are based upon a large 
number of experiments on bearings in which the races were made 
of a good quality of hardened steel: 

1. For a flat or conical raceway having three or four points of 
contact the value of k varies from 420 to 700. 

2. For curved raceways whose radius of curvature equals % d 
and having two-point contact the value of k is 1,400. 

3. For special races and balls made of special alloy steel the 
above values may be increased 50 per cent. 

(6) Load per bearing. — According to Art. 385(a), the total load 
upon a ball bearing may be assumed as being supported by one- 
fifth of the number of baUs in the bearing, hence multiplying 

the values of w by — we get the total load 
5 

W = ^ (631) 



in which Z denotes the number of balls in the bearing. 

(c) Crushing strength of balls. — In Table 103 are given the 
approximate crushing strengths of the commercial sizes of 
regular tool steel balls. These values according to R. H. Grant 
are considered reliable, and were adopted by the manufacturers 

Table 103. — Crushing Strength of Tool-steel Balls 



Diam. of 


Ultimate 


Diam. of 


Ultimate 


Diam. of 


Ultimate 


ball 


strength, lb. 


ball 


strength, lb. 


ball 


strength, lb . 


He 


390 


% 


14,000 


l He 


88,000 


%2 


875 


Vie 


19,100 


1 


100,000 


%4 


1,562 


y 2 


25,000 


1H 


125,000 


v 8 


2,450 


He 


31,500 


1H 


156,000 


He 


3,496 


5 A 


39,000 


i« 


225,000 


V32 


4,780 


h 


56,250 


i« 


263,000 


H 


6,215 


13 Ae 


66,000 


IK 


306,000 


He 


9,940 


y s 


76,000 


2 


400,000 



576 BALL BEARING DATA [Chap. XX 

after several years of testing. Data pertaining to special alloy- 
steel balls are not available, but it is safe to assume that the 
crushing loads will exceed those given in Table 103 by 25 to 
50 per cent. According to Grant a factor of safety of ten should 
be used in selecting balls for bearings. 

394. Coefficient of Friction. — In order to compare ball bearings 
with ordinary plain bearings, the coefficient of friction is referred 
to the diameter of the shaft. Stribeck found experimentally 
that the coefficient of friction of a good radial ball bearing having 
curved raceways is independent of the speed within wide limits 
and has an average value of 0.0015. This coefficient will prac- 
tically be double this value when the load on the bearing is re- 
duced to approximately one-tenth of the maximum load. The 
magnitude of the coefficient of friction in a radial bearing will 
also depend upon the axial thrust coming upon it. According 
to some experimental data published in the American Machinist 
of March, 1909, the coefficient of friction for a radial ball bearing 
subjected to a constant radial load and a variable axial thrust 
increased from 0.004 at a speed of 200 revolutions per minute 
to 0.012 at a speed of 1,200 revolutions per minute. 

395. Ball Bearing Data.— Through the efforts of the Com- 
mittee on Standards appointed by the Society of Automobile 
Engineers, practically all types of radial ball bearings have been 
standardized. In a report submitted to the society at the Spring 
meeting in 1911 were included tables giving standard dimensions 
of light, medium, and heavy radial bearings. Some manufacturers 
make a fourth series known as the extra heavy. According to the 
catalogs of the various prominent manufacturers thrust bearings 
are made in light, medium, and heavy series. With few excep- 
tions the ball bearing manufacturers have adopted the English 
unit for the ball dimensions and the metric unit for the remaining 
dimensions of the bearing. 

(a) Hess-Bright radial bearings.— In Table 104 are given the 
leading dimensions of the light, medium, and heavy series of. the 
wide-type Hess-Bright radial bearings. The symbols denoting 
the dimensions refer to the key drawing of Fig. 325 (c) . The load- 
carrying capacity as given in Table 104 is based on a steady load 
and a constant speed not exceeding 200 revolutions per minute. 
The load rating of any size bearing operating at any given speed 
not exceeding 1,500 revolutions per minute may be determined 



Art. 395] H ESS-BRIGHT RADIAL BEARINGS 577 

Table 104. — Data Pertaining to Hess-Bright Radial Bearings 



No. and type 
of 


Dimensions in 


mm. 


Diam. of 
of 


Capacity 








at 


bearing 


1 


2 


3 


balls 


200 r.p.m. 




200 


30 


10 


9 


He 


130 




201 


32 


12 


10 


He 


145 




202 


35 


15 


11 


He 


165 




203 


40 


17 


12 


VZ2 


240 




204 


47 


20 


14 


y± 


350 




205 


52 


25 


15 


8. 


395 




206 


62 


30 


16 


550 




207 


72 


35 


17 


He 


660 


S 


208 


80 


40 


18 


H 


860 


"C 


209 


85 


45 


19 


% 


945 


m 


210 


90 


50 


20 


% 


990 


*3 


211 


100 


55 


21 


He 


1,255 


!' 


212 


110 


60 


22 


H 


1,630 


213 


120 


65 


23 


Yi 


1,760 




214 


125 


70 


24 


He 


1,870 




215 


130 


75 


25 


2,200 




216 


140 


80 


26 


% 


2,750 




217 


150 


85 


28 


x He 


3,080 




218 


160 


90 


30 


H 


3,630 




219 


170 


95 


32 


H 


3,850 




220 


180 


100 


34 


13 Ae 


4,180 




221 


190 


105 


36 


14, 


4,840 




222 


200 


110 


38 


li 


5,280 




300 


35 


10 


11 


A 


220 




301 


37 


12 


12 


y* 


265 




302 


42 


15 


13 


y± 


285 




303 


47 


17 


14 


He 


395 




304 


52 


20 


15 


He 


440 




305 


62 


25 


17 




660 




306 


72 


30 


19 


He 


880 




307 


80 


35 


21 


H 


1,100 


.Si 


308 


90 


40 


23 


He 


1,430 


cu 


309 


100 


45 


25 


H 


1,760 


to 


310 


110 


50 


27 


1 He 


2,090 


a 


311 


120 


55 


29 


H 
l He 


2,530 


s 


312 


130 


60 


31 


2,970 


t5 


313 


140 


65 


33 


Vs 


3,410 


1 


314 


150 


70 


35 


x He 


3,895 


315 


160 


75 


37 


1 


4,400 




316 


170 


80 


39 


IKe 


4,995 




317 


LS0 


85 


41 


IX 


5,500 




318 


190 


90 


43 


lHe 


6,160 




319 


200 


95 


45 


IK 


6,820 




320 


215 


100 


47 


lHe 


7,435 




321 


225 


105 


49 


m 


8,140 




322 


240 


110 


50 


IK 


9,680 




403 


62 


17 


17 


y 2 


770 




404 


72 


20 


19 


He 


1,145 




405 


80 


25 


21 


% 


1,385 


• 


406 


90 


30 


23 


1 He 


1,650 




407 


100 


35 


25 


3 A 
l He 


1,980 




408 


110 


40 


27 


2,310 




409 


120 


45 


29 


y% 


2,640 


S 


410 


130 


50 


31 


16 Ae 


3,080 


'S 


411 


140 


55 


33 


1 


3,465 


to 


412 


150 


60 


35 


IKe 


4,400 


>, 


413 


160 


65 


37 


m 


4,950 


1 


414 


180 


70 


42 


m 


6,095 


53 


415 


190 


75 


45 


lHe 


6,710 


M 


416 


200 


80 


48 


1% 


7,370 




417 


210 


85 


52 


We 


8,030 




418 


225 


90 


54 


1%6 


9,460 




419 


250 


95 


55 


1^6 


10,390 




420 


265 


100 


60 


1^6 


12,650 




421 


290 


105 


65 


m 


13,500 




422 


320 


110 


70 


2 


15,400 



578 



S. K. F. RADIAL BEARINGS 



[Chap. XX 



by multiplying the capacity given in the table by the speed 
coefficient obtained from Fig. 335. The graph of Fig. 335 is 
based upon information published in the trade literature issued 
by the Hess-Bright Mfg. Co. 

(6) S. K. F. radial bearing. — The leading dimensions and load 
rating of the light, medium, and heavy series of the narrow 
type of S.K.F. self-aligning radial bearing, similar to that shown 
in Fig. 325(d), are given in Table 105. The load rating given in 



0.8 



</) 0.6 



0.5 



±K 



^ 



500 
Revolu-Hons per minu-fe 

Fig. 335. 



1000 



this table applies to a steady load and a constant speed not 
exceeding 300 revolutions per minute. To determine the per- 
missible load capacity for a bearing running at other speeds than 
300, the rating given in the table must be multiplied by the speed 
coefficient obtained from the graph of Fig. 336. This graph was 
plotted from data deduced from the load-carrying capacities and 
speeds given in the trade publication issued by the S. K. F. 
Ball Bearing Co. 

(c) F. and S. thrust bearing. — The dimensions and load ratings 
given in Table 106 pertain to the light, medium, and heavy series 
of F. and S. spherical seated type of ball bearing shown in Fig. 



Art. 395] S. K. F. RADIAL BEARINGS 579 

Table 105. — Data Pertaining to S. K. F. Radial Bearings 



No. and type 


Dimensions in mm. 


Capacity 




01 












at 


bearing 


1 


2 


3 


4 


5 


300 r.p.m. 




1,200 


30 


10 


9 






200 




1,201 


32 


12 


10 










220 




1,202 


35 


15 


11 










285 




1,203 


40 


17 


12 










365 




1,204 


47 


20 


14 










465 




1,205 


52 


25 


15 










630 




1,206 


62 


30 


16 










850 




1,207 


72 


35 


17 


2 








970 


_<B 


1,208 


80 


40 


18 


2 








1,210 


<3 


1,209 


85 


45 


19 


2 








1,400 


to 


1,210 


90 


50 


20 


2 








1,575 


J3 


1,211 


100 


55 


21 


2 








1,930 


SP 


1,212 


110 


60 


22 


2 








2,205 


u 


1,213 


120 


65 


23 


2 








2,430 




1,214 


125 


70 


24 


2 








2,810 




1,215 


130 


75 


■ 25 


2 








2,980 




1,216 


140 


80 


26 


3 








3,310 




1,217 


150 


85 


28 


3 








4,080 




1,218 


160 


90 


30 


3 








4,630 




1,219 


170 


95 


32 


3 








5,520 




1,220 


180 


100 


34 


3 








6,060 




1,221 


190 


105 


36 


3 








7,060 




1,222 


200 


110 


38 


3 








7,720 




1,300 


35 


10 


11 










265 




1,301 


37 


12 


12 










350 




1,302 


42 


15 


13 










385 




1,303 


47 


17 


14 










550 




1,304 


52 


20 


15 










575 




1,305 


62 


25 


17 










885 




1,306 


72 


30 


19 


2 








1,100 




1,307 


80 


35 


21 


2 








1,430 


.2 


1,308 


90 


40 


23 


2 








1,760 





1,309 


100 


45 


25 


2 








2,200 


m 


1,310 


110 


50 


27 


2 








2,540 


B 


1,311 


120 


55 


29 


2 








3,310 


p 


1,312 


130 


60 


31 


2 








3,860 


-a 


1,313 


140 


65 


33 


3 








4,410 


1 


1,314 


150 


70 


35 


3 








5,070 




1,315 


160 


75 


37 


3 








5,850 




1,316 


170 


80 


39 


3 








6,070 




1,317 


180 


85 


41 


3 


42.0 


8,720 




1,318 


190 


90 


43 


3 


44.0 


9,100 




1;319 


200 


95 


45 


3 


46.0 


10,900 




1,320 


215 


100 


47 


3 


58.0 


11,400 




1,321 


225 


105 


49 


3 


60.0 


13,200 




1,322 


240 


110 


50 


3 


64.0 


14,300 




402 


52 


15 


15 


1 




630 




403 


62 


17 


17 


1 








885 




404 


72 


20 


19 


2 








1,145 




405 


80 


25 


21 


2 








1,435 




406 


90 


30 


23 


2 








1,765 




407 


100 


35 


25 


2 








2,205 




408 


110 


40 


27 


2 








2,535 


.2 


409 


120 


45 


29 


2 








3,300 


<3 


410 


130 


50 


31 


2 








3,850 


m 


411 


140 


55 


33 


3 








4,410 


>> 


412 


150 


60 


35 


3 








5,070 


£3 


413 


160 


65 


37 


3 








6,060 


a 


414 


180 


70 


42 


3 








8,820 


415 


190 


75 


45 


3 








9,080 




416 


200 


80 


48 


3 








11,000 




417 


210 


85 


52 


3 


54.5 


11,000 




418 


225 


90 


54 


3 


60.0 


13,250 




419 


250 


95 


55 


3 


66.0 


15,100 




420 


265 


100 


60 


3 


68.5 


16,550 



580 



F. AND S. THRUST^BEARINGS 



[Chap. XX 



330(6). Using the load capacity as given in the table as a basis, 
the permissible maximum rating for a bearing running at any- 
constant speed in excess of 50 revolutions per minute may be 



1.2 



100 



0.9 



•£ 0.8 

O 

o 



0.7 



0.6 



0.5 



0.4 



0.3 



200 



Revolu-Hons per minu-He 
300 400 500 



1000 



1500 



r 

-' 

\ 

\ 


■ — ■ ■ -H 

J 1. 

Yfff 



4000 



3500 



3000 



2500 



2000 



1500 



Revolutions per minu+e 
Fig. 336. 



determined by multiplying the tabular value by the speed coef- 
ficient obtained from Fig. 337. The graph of Fig. 337 is based 
upon the load capacities given in the trade literature issued by 



Art. 395] 



F. AND S. THRUST BEARINGS 



581 



Table 106. — Data Pertaining to F. & S. Spherical Seated Thrust 
Bearings 



.. No. and type 




Di 


tnensions in mm. 




Balls 


Capacity 


of 




















at 


bearing 


1 


2 


3 


4 


5 


6 


No. 


Diam. 


50 r.p.m. 




AJL 10 


30 


10 


14 


12 


21 


21 


8 


8 


675 




AJL 15 


35 


15 


16 


16 


24 


25 


10 


850 




AJL 20 


40 


20 


16 


21 


26 


30 


12 


g 


1,000 




AJL 25 


45 


25 


16 


26 


33 


35 


14 


1,200 




AJL 30 


53 


30 


18 


32 


38 


40 


16 


A 


1,350 




AJL 35 


62 


35 


21 


37 


44 


50 


16 


He 


2,100 




AJL 40 


64 


40 


21 


42 


49 


50 


17 


S s 


2,250 


S 


AJL 45 


73 


45 


25 


47 


55 


60 


16 


3,000 


"S 


AJL 50 


78 


50 


25 


52 


60 


65 


18 


ru 


3,400 


to 


AJL 55 


88 


55 


28 


57 


65 


70 


17 


4,400 


s? 


AJL 60 


90 


60 


28 


62 


70 


75 


18 


He 


4,650 


AJL 65 


100 


65 


32 


67 


75 


80 


18 


y 2 


6,100 


13 


AJL 70 


103 


70 


32 


72 


80 


85 


19 


y 2 


6,400 




AJL 75 


110 


75 


32 


77 


85 


90 


20 


y 2 


6,700 




AJL 80 


115 


80 


35 


82 


90 


95 


21 


b 


7,100 




AJL 85 


125 


85 


38 


88 


97 


105 


18 


9,500 




AJL 90 


132 


90 


39 


93 


103 


110 


19 


% 


10,000 




AJL 95 


140 


95 


41 


98 


109 


115 


18 


%: 


11,400 




AJL 100 


148 


100 


42 


103 


118 


120 


19 


12,000 




AJL 105 


155 


105 


43 


110 


130 


130 


18 


K 


13,600 




AJL 110 


160 


110 


43 


115 


135 


135 


19 


A 


14,400 




BJL 25 


52 


25 


19 


26 


40 


40 


13 


He 


1,700 




BJL 30 


60 


30 


21 


32 


45 


45 


13 


1 


2,450 




BJL 35 


68 


35 


24 


37 


55 


55 


13 


3,350 




BJL 40 


76 


40 


27 


42 


60 


60 


13 


4,400 




BJL 45 


85 


45 


30 


47 


65 


65 


13 


He 


5,500 




BJL 50 


92 


50 


33 


52 


75 


75 


13 


% 


6,800 


m 


BJL 55 


100 


55 


35 


57 


80 


80 


13 


% 


7,500 


'£ 


BJL 60 


106 


60 


37 


62 


85 


85 


13 


% 


8,250 


s 


BJL 65 


112 


65 


38 


67 


90 


90 


14 


9,700 


a 

.2 


BJL 70 


120 


70 


40 


72 


95 


95 


14 


&. 


10,600 


BJL 75 


128 


75 


43 


77 


105 


105 


14 


12,400 


■3 


BJL 80 


136 


80 


46 


82 


110 


110 


14 


v% 


14,400 


i 


BJL 85 


145 


85 


49 


88 


120 


120 


14 


ls Ae 


16,500 


BJL 90 


155 


90 


52 


93 


125 


125 


14 


1 


18,800 




BJL 95 


165 


95 


56 


98 


130 


130 


14 


lMe 


21,300 




BJL 100 


172 


100 


59 


103 


140 


140 


14 


iy s 


24,000 




BJL 105 


180 


105 


62 


110 


145 


145 


14 


IHe 


26,500 




BJL 110 


190 


110 


65 


115 


155 


155 


14 


lJi 


29,500 




BJL 115 


200 


115 


68 


120 


160 


160 


14 


IHe 


32,500 




BJL 120 


210 


120 


72 


125 


170 


170 


14 


m 


35,500 




CJL 20 


50 


20 


21 


21 


35 


35 


10 


% 


1,900 




CJL 25 


60 


25 


25 


26 


45 


45 


10 


1S A 2 


3,000 




CJL 30 


73 


30 


30 


32 


50 


50 


10 


He 


4,300 




CJL 35 


80 


35 


33 


37 


60 


60 


10 


% 


5,250 




CJL 40 


90 


40 


38 


42 


65 


65 


10 


% 


7,000 




CJL 45 


100 


45 


42 


47 


75 


75 


10 


ls Ae 


8,900 




CJL 50 


110 


50 


47 


52 


80 


80 


10 


2 %2 


11,000 


s 


CJL 55 


120 


55 


52 


57 


90 


90 


10 


1 


13,500 


'S 


CJL 60 


130 


60 


56 


62 


95 


95 


10 


We 


15,200 


m 


CJL 65 


140 


65 


61 


67 


105 


105 


10 


We 


19,000 


>> 


CJL 70 


150 


70 


65 


72 


110 


110 


10 


IK 


21,000 


> 


CJL 75 


160 


75 


70 


77 


120 


120 


10 


1% 


25,500 


s 


CJL 80 


170 


80 


74 


82 


125 


125 


10 


l 7 Ae 


28,000 


B 


CJL 85 


180 


85 


78 


88 


135 


135 


10 


1M 


30,000 




CJL 90 


190 


90 


83 


93 


140 


140 


10 


m 


35,500 




CJL 95 


195 


95 


86 


98 


145 


145 


10 


i 1 K 6 


38,500 




CJL 100 


205 


100 


89 


103 


155 


155 


10 


m 

± 13 Ae 
l 13 Ae 


41,000 




CJL 115 


220 


115 


93 


120 


170 


170 


11 


48,500 




CJL 130 


240 


130 


95 


135 


185 


185 


12 


53,000 




CJL 150 


260 


150 


99 


155 


205 


205 


13 


W 


61,500 



582 



F. AND S. THRUST BEARINGS 



[Chap. XX 



1.0 



0.9 



0.8 



0.7 



0.6 



0.5 



(J) 



0.4 



0.3 



0.2 



•0.1 



Revolu+ions per minu+e 
100 200 300 400 



500 




2000 



1500 



1000 



500 



Revolu+ions per minu+e 
Fig. 337. 



Art. 395] GURNEY RADIO-THRUST BEARINGS 583 

The Bearings Co. of America, the distributors of the F. and S. 
bearings. 

(d) Gurney radio-thrust bearing. — The Gurney radio-thrust 
bearing is manufactured in the following three standard types: 

1. Type RT having a thrust capacity equivalent to 100 per 
cent, of the rated radial load. 

2. Type RT 150 having a thrust capacity equivalent to 150 per 
cent, of the rated radial load. 

3. Type RT 200 having a thrust capacity equivalent to 200 per 
cent, of the rated radial load. 

In Table 107 are given the leading dimensions, load-carrying 
capacity, and speed rating for the light, medium, and heavy series 
of the RT type Gurney radio-thrust bearing similar to that shown 
in Fig. 333. It should be understood that the ratings given in 
Table 107 are not intended for all conditions of operation, but 
that they apply only to the class of service in which uniform 
load and constant speed prevail. Furthermore, the speed rating 
applies to the type of mounting in which the inner race rotates. 
When the outer race revolves, the permissible speed should be 
taken as 60 per cent, of that listed in the table. According to 
information furnished by the manufacturer, the rated capacity 
of the RT 150 type is 95 per cent, and that of the RT 200 type is 
90 per cent, of the rating given in Table 107. When the speed 
of the inner race is above or below the value given in the table, 
the permissible load capacity is obtained by multiplying the rated 
load by the so-called load factor. This factor depends upon the 
speed coefficient and may be determined from the graph of Fig. 
338. By the term speed coefficient is meant the ratio of the given 
revolutions per minute to the rated revolutions per minute given 
in the table. 

To determine the capacity of a Gurney radio-thrust bearing 
having given the radial load coming upon it, the following rule 
used by the Gurney Ball Bearing Co. is recommended: 

Rule I. — "Subtract the actual radial load from the rated load 
and multiply the remainder by the thrust percentage of the bearing." 

If it is required to determine the available radial load capacity 
of a radio-thrust bearing having given the axial thrust, the 
following rule should be used: 

Rule II. — "Divide the actual thrust by the thrust percentage 
of the bearing and subtract the result from the rated load." 



584 



GURNEY RADIO-THRUST BEARINGS [Chap. XX 



The use of the various factors and rules just given is shown 
best by applying them to a problem, as follows: *j ■ 



Speed CoefTicien + 
0.5 1.0 1.5 



2.5 




4.0 3.5 

Speed Coef-Ficien-H 
Fig. 338. 



2.5 



Problem. — It is required to determine the thrust capacity of No. 310 RT 
150 radio-thrust bearing running at 750 revolutions per minute, assuming 
that the bearing is carrying a radial load of 1,000 pounds. 

Solution.— The load rating of bearing No. 310 RT 150 is 3,000 X 0.95 or 
2,850 pounds at 625 revolutions per minute. From Fig. 338 the load factor 

750 
corresponding to a speed factor of w^z, or 1.2, is 0.91, hence the load capacity 

at 750 revolutions per minute is 2,850 X 0.91 or 2,590 pounds. Applying 
the first rule given above, the thrust capacity of the given bearing operating 
under the above conditions is 1.5 (2,590 — 1,000) or 2,385 pounds. 



Art. 395] GURNEY RADIO-THRUST BEARINGS 585 

Table 107. — Data Pertaininq to Gurney Radio-thrust Bearings 









Dimensions 




Balls 






No. an 


I type 














Load 


Speed 
















OI 

bearing 


1 


2 


3 


4 


No. 


Diam., 


rating 


rating 






mm. 


mm. 


mm. 


in. 




in. 








204 RT 


47 


20 


14 


Hi 


13 


%2 


480 


1,450 




205 RT 


52 


25 


15 


Ha 


14 


%2 


530 


1,250 




206 RT 


62 


30 


16 


% 


15 


He 


720 


1,050 




207 RT 


72 


35 


17 


16 


¥s 


1,100 


900 




208 RT 


80 


40 


18 


He 


16 


He 


1,480 


800 




209 RT 


85 


45 


19 


He 


17 


He 


1,570 


740 


3 


210 RT 


90 


50 


20 


H.A 


18 


He 


1,670 


680 


■fl 


211 RT 


100 


55 


21 


%4 


18 


X 


1960 


600 


m 


212 RT 


110 


60 


22 


Ha 


17 


He 


2,570 


550 


-£ 


213 RT 


120 


65 


23 


Ha 


17 


H 


3,230 


500 


bD 


214 RT 


125 


70 


24 


Ha, 


18 


% 


3,420 


475 


h5 


215 RT 


130 


75 


25 


Ha 


19 


X 


3,610 


450 




216 RT 


140 


80 


26 


¥%2 


19 


1 He 


4,370 


420 




217 RT 


150 


85 


28 


¥i2 


18 


¥a 

lz He 


4,850 


390 




218 RT 


160 


90 


30 


%2 


18 


5,700 


370 




219 RT 


170 


95 


32 


X 


18 


H 


6,650 


350 




220 RT 


180 


100 


34 


X 


17 


™Ae 


7,220 


330 




221 RT 


190 


105 


36 


X 


17 


1 


8,080 


315 




222 RT 


200 


110 


38 


X 


17 


1Mb 


9,310 


300 




304 RT 


52 


20 


15 


Ha 


13 


He 


620 


1.200 




305 RT 


62 


25 


17 


He 


13 


X 


860 


1,050 




306 RT 


72 


30 


19 


8: 


13 


He 


1,190 


950 




307 RT 


80 


35 


21 


13 


X 


1,570 


850 




308 RT 


90 


40 


23 


Ha 


13 


He 


2,000 


750 




309 RT 


100 


45 


25 


Ha 


13 


X 


2,470 


675 


m 


310 RT 


110 


50 


27 


Ha 


13 


\Xe 


3,000 


625 


■S 


311 RT 


120 


55 


29 


¥z2 


13 




3,570 


575 


n 


312 RT 


130 


60 


31 


%2 


14 


Aie 


4,470 


525 


s 


313 RT 


140 


65 


33 


%2 


14 


X 


5,230 


475 


314 RT 


150 


70 


35 


¥z2 


14 


x He 


5,990 


450 




315 RT 


160 


75 


37 


X 


14 


1 


6,750 


425 


i 


316 RT 


170 


80 


39 


X 


14 


1Kb 


7,600 


400 


317 RT 


180 


85 


41 


X 


14 


IX 


8,550 


375 




318 RT 


190 


90 


43 


X 


14 


1Kb 


9,590 


350 




319 RT 


200 


95 


45 


X 


14 


IJi 


10,640 


330 




320 RT 


215 


100 


47 


X 


14 


m> 


11,740 


315 




321 RT 


225 


105 


49 


X 


14 


12,830 


300 




322 RT 


240 


110 


50 


X 


14 


IK 


15,200 


285 




404 RT 


72 


20 


19 


He 


9 


He 


1,380 


1,250 




405 RT 


80 


25 


21 


He 


9 


X 


1,620 


1,075 




406 RT 


90 


30 


23 


Ha 


10 


x He 


2,280 


875 




407 RT 


100 


35 


25 


Ha 


10 


¥a 


2,660 


775 




408 RT 


110 


40 


27 


Ha 




13 He 


3,520 


700 




409 RT 


120 


45 


29 


Ha 




X 


4,090 


650 




410 RT 


130 


50 


31 


¥z2 




15 Ae 


4,700 


600 


.2 


411 RT 


140 


55 


33 


II 2 

%2 




1 


5,320 


550 


g 


412 RT 


150 


60 


35 




1Kb 


5,990 


500 


to 


413 RT 


160 


65 


37 


¥Z2 




IX 


6,750 


460 


>> 


414 RT 


180 


70 


42 


¥%2 




IHe 


9,190 


425 


03 


415 RT 


190 


75 


45 


X 




IX 


10,070 


400 


w 


416 RT 


200 


80 


48 


X 




IHe 


11,020 


375 


417 RT 


210 


85 


52 


X 




IX 


11,970 


350 




418 RT 


225 


90 


54 


X 




IX 


14,060 


325 




419 RT 


250 


95 


55 


X 




\¥a 
IX 


16,340 


300 




420 RT 


265 


100 


60 


X 




18,810 


285 




421 RT 


290 


105 


65 


X 




2X 


23,130 


270 




422 RT 


320 


110 


70 


X 




2He 


28,500 


250 



Problem. — It is required to determine the radial load capacity of bearing 
No. 310 RT 150 running at 750 revolutions per minute and carrying an axial 
load of 2,100 pounds. 

Solution. — Applying Rule II, we find that the axial load of 2,100 pounds is 

equivalent to ' or 1,400 pounds. According to the preceding problem 



586 



MOUNTING BALL BEARINGS 



[Chap. XX 



the load capacity of the given size of bearing running at 750 revolutions per 
minute is 2,590 pounds, whence the magnitude of the radial load that may be 
placed upon the bearing in addition to the 2,100 pounds axial load is 
2,590 - 1,400 or 1,190 pounds. 

396. Mounting Ball Bearings. — (a) Radial bearings. — In 
general, the requirements of a correct mounting for radial 
ball bearings carrying no axial thrust are as follows. 

1. The shaft and sleeves or losses of pulleys, sprockets, and 
gears upon which the inner races of radial bearings are to be 
mounted must be turned and ground accurately, and the hous- 
ings into which the outer races are fitted must be bored true so 
as to insure the concentric running of the races. 

2. In some installations, as for example on a line- or counter- 
shaft, it is impossible to provide the inner race with a driving 





Fig. 339. 



fit on the shaft and to clamp it against a shoulder. In such 
cases a device called an adapter is used. The adapter is nothing 
more than a split conical sleeve fitted over the shaft and pro- 
vided with a nut and nut-lock as shown in Fig. 339. It is evi- 
dent that the inner race may be rigidly clamped to the shaft 
at any desired position. 

3. Whenever it is necessary to use a split housing the parts 
must be fitted together correctly so that the inner race will not 
become distorted due to any clamping action of the housing. 

4. The inner race of the bearing must be retained in a fixed 
position. To this end it is made with a tight fit on the shaft and 
is held rigidly in position against a shoulder on the shaft by 
means of a nut and nut-lock. In Figs. 170 and 171 are shown 
typical mountings of radial bearings. 



Art. 396] MOUNTING BALL BEARINGS 587 

5. The outer race should readily take up a free position rela- 
tive to the balls and inner race, thus insuring a more nearly 
perfect distribution of the load over the entire outer race. To 
permit a slight degree of axial movement, the outer race should 
have a so-called "sucking fit" in the housing and should never 
be held in place rigidly. 

6. The mounting must be so designed that the ball bearing will 
not be left exposed to the action of water, dust, grit, and other 
foreign matter. Provision must also be made for retaining the 
lubricant. Various forms of caps for closing the sides of the hous- 
ing are used, some of which are shown in Figs. 170, 194, 195, and 
339. Experience has demonstrated that one or two cored grooves 
in the caps packed with a stiff grease or felt are effective in keep- 
ing out grit and water, and in preventing the escape of the lubri- 
cant. The grooves are generally made from Y± to % inch deep, 
and the bore through which the shaft passes must be made ap- 
proximately 3^4 inch larger than the shaft. 

7. Whenever a shaft having no thrust bearing is supported by 
several radial bearings, satisfactory results are assured by secur- 
ing the outer race of one of these bearings against axial move- 
ment, while the outer races of the remaining bearings must be 
left free to locate themselves. 

(b) Single-thrust bearings. — In mounting one-direction thrust 
bearings the rotating race must be pressed against a suitable 
shoulder on the shaft by a light driving fit. The shoulder on the 
shaft must be of sufficient height so as not to subject the rotating 
race to an undue bending action. If it is impossible to provide a 
proper shoulder, a suitable washer or sleeve must be used between 
the shoulder and the race. The bore of the stationary race is 
made considerably larger than that of the rotating race. To 
secure satisfactory service with a thrust bearing, the load upon 
the balls must be distributed evenly. For this purpose the sta- 
tionary race is provided with a spherical face, so that the complete 
thrust bearing may be supported on a spherical seated washer. 
The bore of the latter must be large and in the majority of instal- 
lations, this washer should be free to move laterally, thus provid- 
ing for any shaft deflection that is liable to occur. Whenever 
possible the center of the spherical seated race and washer 
should be located at, or near, the center of the radial bearing used 
in conjunction with the thrust bearing. As in the case of radial 
bearings, the mountings of all types of thrust bearings must ex- 



588 MOUNTING BALL BEARINGS [Chap. XX 

elude water, grit, and foreign matter and at the same time retain 
the lubricant. Furthermore, the same degree of workmanship on 
the various parts of the thrust bearing mounting is required as is 
necessary for the radial bearing. 

(c) Double-thrust bearing. — In mounting a double-thrust bear- 
ing of the type illustrated in Fig. 331, the spherical faced 
stationary or outer ball races must bear against accurately ma- 
chined spherical seats in the housing and cap. The central or 
rotating race must be fastened to the shaft and held against a 
suitable shoulder by means of a sleeve and nut-lock. The self- 
contained double-thrust bearing shown in Fig. 331 is mounted 
by fastening the central race to the shaft in a manner similar to 
that described for the plain double-thrust bearing. It is unneces- 
sary to have an accurately bored seat for the outer casing, but 
machined faces must be provided between which the entire bear- 
ing may be clamped. 

(d) Radio-thrust bearing. — The single radio-thrust bearings 
shown in Figs. 325 (&) and 333 can take a thrust in only one direc- 
tion and for that reason some care must be exercised in mounting 
such bearings. The Gurney radio-thrust bearing is made with 
a certain amount of looseness which must be taken up in the 
mounting. As in the case of the radial bearings, the inner race 
is mounted on the shaft with a light press fit and held against a 
suitable shoulder on the shaft by means of a nut and nut-lock. 
The outer race is made a push fit in the housing and must be held 
against a suitable shoulder in the housing or against the end of an 
adjustable cap. Typical mountings of radio-thrust bearings are 
shown in Figs. 171 and 195. 

The double-row bearing shown in Fig. 334 really consists of 
two radio-thrust bearings located within an outer casing. Since 
this type of bearing is capable of supporting a radial load and at 
the same time take a thrust in either direction, the method of 
mounting depends upon the nature of the loading. As in the 
radial bearings, the inner race is fixed to the shaft, and if axial 
loads from both directions must be taken care of, the outer race 
must be clamped rigidly between a suitable shoulder on the 
machine frame and the end of an adjustable cap, as shown in 
Fig. 194. An installation of an ordinary radial bearing used in 
conjunction with a double-row radio-thrust bearing, the latter 
taking a thrust in one direction only, is shown in Fig. 170. 



Art. 396] REFERENCES 589 



References 

Bearings and their Lubrication, by L. P. Alford. 
Handbook for Machine Designers and Draftsmen, by F. A. Halsey. 
Bearings, Trans. A. S. M. E., vol. 27, p. 441. 
Ball Bearings, Trans. A. S. M. E., vol. 29, p. 367. 

Ball Bearings, Paper before Electric Vehicle Assoc, of Amer., Apr. 14, 1915. 
The Design of Ball Bearings, Ind. Eng'g of the Eng'g Digest, vol. 13, pp. 
24, 71, 117. 
A Series of Tests on Roller Bearings, Amer. Mach., vol. 38, p. 769. 
The Friction of Roller Bearings, Mchy., vol. 12, p. 62. 
The Manufacture of Steel Balls, Mchy., vol. 18, p. 590. 
Gurney Ball Bearing Engineering Bulletins, Gurney Ball Bearing Co. 
Ball Bearing Engineering, Hess-Bright Mfg. Co. 
Ball Bearing Applications, The New Departure Mfg. Co. 
Bulletins published by S. K. F. Ball Bearing Co. 



GENERAL INDEX 



Acme thread, 79 
Addendum, 281 
Adjustments for alignment, 526 

for wear, 519 
Akron clutch, 432 
Alco clutch, 433 
Alignment, adjustment for, 526 
Aluminum, 41 

bronze, 41 

copper, 42 
-zinc, 42 

zinc, 42 
Anchors, chain, 220 
Angle and plate connection, double, 
68 

single, 64 
Annealing, 43 
Antifriction curve, 554 
Arms of spur gears, 307 
Auburn thrust ball bearings, 572 
Automatic brakes, 476 

analysis of, 484 
Automobile bolts, 82 
Axial brakes, 472 

Babbitt metal, 42, 515 
Bakelite Micarta-D gears, 306 
Ball, load per, 575 
Balls, crushing strength of, 575 
Ball bearing, Auburn thrust, 472 

Gurney radio-thrust, 573 

Norma, 570 

Radax, 573 

radial and thrust, 573 

S. K. F., 569 
Ball bearings, coefficient of friction 
of, 576 

data for, 576 

duplex, 574 

mounting of, 586 

pressures on, 575 

radial, 569 

thrust, 570 



Ball raceways, forms of, 566 
Band brakes, 469 

clutch, 454 

analysis of a, 456 
Bar, eccentrically loaded, 12 

straight prismatic, 12 
Barth key, 113 

Beams, end connections for, 66 
Bearing, Auburn thrust ball, 572 

bolts, design of, 534 

caps, design of, 534 

DeLaval thrust, 544 

design, formulas for, 532 

friction, coefficient of, 530 

four-part, 520 

Gurney radio-thrust ball, 573 

Hyatt, 559 

materials, 514 

Mossberg roller, 556 

Norma ball, 570 
roller, 557 

pintle, 546 

pressures, 527 

on power screws, 108 
table of, 528 

Radax ball, 573 

radial roller, 556 

S. K. F. ball, 569 

split, 519 

Timken roller, 559 

with thrust washers, 543 
Bearings, capacities of Norma, 563 

collar thrust, 543 

conical radial roller, 558 
roller thrust, 561 

connecting rod, 523 

cylindrical roller thrust, 560 

data for ball, 576 
for roller, 562 

dimensions of Hyatt, 564 
of Norma, 563 

duplex ball, 574 

flexible roller, 559 



591 



592 



GENERAL INDEX 



Bearings, for friction gearing, 277 
for radial and axial loads, 545 
friction of collar thrust, 550 
journal, 513 
length of, 529 
marine thrust, 543 
mounting of ball, 586 

of roller, 564 
multiple disc step, 547 
pressures on ball, 575 
proportions of common split, 
540 
of journal, 539 
of pedestal, 540 
of post, 540 
radial and thrust ball, 573 

ball, 569 
radiating capacity of, 530 
right line, 513 
single disc step, 546 
sliding, 513 
• solid, 520 
step, 546 

table of length of, 529 
temperature of, 533 
thrust, 513 
ball, 570 
Tower's experiments on collar 
thrust, 551 
on step, 552 
work lost in collar thrust, 550 
in conical journal, 537 
in cylindrical journal, 535 
in Schiele pivofc, 553 
Becker's brake, 477 
analysis of, 477 
Belt, block type of V, 171 
chain type of V, 172 
fastenings, tests of, 155 
ratio of tensions, 156 
selection of size, 160 
tandem transmission, 162 
Belts, tension in, 155 
Belting, analysis of V, 172 

coefficient of friction for, 159 
experiments on steel, 152 
leather, 147 
rubber, 148 
steel, 151 



Belting, strength of leather, 149 
of rubber, 150 
Taylor's experiments on, 161 
textile, 150 

working stresses for leather, 160 
Bending moments on shafts, 500 

stresses in wire rope, 196 
Bessemer process, 34 
Be ve] -friction gearing, 266 
Bevel gear teeth, form of, 323 
Bevel gears, acute-angle, 326 
advantages of spiral, 345 
arms of, 339 
bearing pressures due to, 338 

due to spiral, 346 
disadvantages of spiral, 345 
Fabroil, 341 
mounting, 341 
obtuse-angle, 328 
resultant pressure on, 336 
right-angle, 330 
skew, 349 
spiral, 344 
strength of cast, 331 

of cut, 333 
tests on, 348 
thrusts due to, 338 
due to spiral, 346 
Birnie's formula, 131 
Billings and Spencer drop hammer, 

262 
Block brakes, 463 

analysis of, 466 
graphical analysis of double, 
468 
chain, 239 

selection of, 242 
sprockets for, 240 
table of Diamond, 240 
clutches, 444 
analysis of, 446 
Bocorselski's universal joint, 391 
Boiler brace, diagonal, 73 
joint, analysis of, 55 
design of, 59 
Bolts, automobile, 82 
carriage, 81 
coupling, 81 
design of bearing, 534 



GENERAL INDEX 



593 



Bolts, machine, 80 

patch, 88 

stay, 89 
Bolts and nuts, U. S. Standard, 78 
Brace, diagonal boiler, 73 
Brake, analysis of automatic, 484 
of Becker's, 477 

Becker's, 477 

cam, 483 

case, 480 

coil, 482 

crane disc, 478 

force analysis of a disc, 475 

graphical analysis of a block, 
468 

Liider's, 476 

Niles, 478 

Pawlings and Harnischfeger, 
479 
Brakes, analysis of block, 466 

automatic, 476 

axial, 472 

band, 469 

block, 463 

clutch, 423 

conical, 472 

differential band, 471 

disc, 473 

disposal of heat in, 487 

double-block, 463 

mechanical load, 476 

post, 463 

simple band, 469 

single-block, 463 
Brass, cast, 39 

white, 43 

wrought, 39 
Bronze, aluminum, 41 

commercial, 40 

manganese, 40 

phosphor, 40 
Bronzes for bearings, 515 
Butt joints, 51 
Buttressed tooth, 314 

Cadillac clutch, 412 
Cam brake, 483 
Cap screws, table of, 84 
Carriage bolts, 81 



Case brake, 480 
Case-hardening, 44 
Castellated nut, 93 
Casting, chilled, 29 

malleable, 29 
Cast iron, 26 

for bearings, 516 
vanadium, 27 
Cementation process, 35 
Chain anchors, 220 
block, 239 

analysis of, 225 
closed joint, 230 
coil hoisting, 216 
conveyor, 228 
Conventry silent, 247 
design data for Morse, 252 
detachable, 228 
drums, 218 
length of roller, 246 
Link Belt silent, 249 
lubrication for bearings, 517 
Morse silent, 250 

proportions of sprockets for 
conveyor, 238 
proportions of sprockets for 

Link Belt silent, 256 
relation between effort and 

load for, 224 
roller, 242 

selection of block, 242 
sheaves, plain, 221 
pocket type, 222 
silent, 247 
sprockets for detachable, 234 

for silent, 251 
strength of closed joint, 232 
of detachable, 229 
of silent, 250 
stud link, 217 
table of, 217 

of Diamond block, 240 

roller, 244 
of Ewart, 230 
of Jeffrey-Mey-Obern, 232 
of Link Belt "400" class, 
232 
silent, 254 
of Union steel, 233 



594 



GENERAL INDEX 



Chain, table of Whitney silent, 253 

Whitney silent, 248 
Chilled casting, 29 
Chrome steel, 37 
Chromium-nickel steel, 37 

-vanadium steel, 38 
Clamp coupling, 386 

dimensions of, 388 
Clark coupling, 400 
Clavarino's. formula, 132 
Clutch, Akron, 432 

Alco, 433 

analysis of a band, 456 
of a block, 446 
of a disc, 434 
of a double-cone, 419 
of the Hele-Shaw, 440 
of a jaw, 402 
of a single cone, 413 
of a split-ring, 451 

asbestos fabric faced cone, 416 

band, 454 

block, 444 

brakes, 423 

cone with cork inserts, 417 

design constants for disc, 437 

Dodge disc, 433 

double cone, 412 

E. G. I., 426 

engaging device for cone, 421 
mechanisms, 458 

experiments on a cone, 418 

face angle of cone, 417 

Ewart block, 444 

Farrel band, 454 

Hele-Shaw, 439 

Horton roller, 457 

Hunter block, 445 

hydraulicaUy operated disc, 427 

Ideal multi-cone, 441 

Johnson, 451 

Knox disc, 423 

leather faced cone, 416 

Litchfield band, 455 

machine tool block, 446 
split-ring, 449 

materials for friction, 406 

Medart block, 445 

Metten disc, 427 



Clutch, Moore and White disc, 442 
motor car cone, 411 
National cone, 412 
Pathfinder disc, 434 
Plamondon disc, 426 
positive, 402 

requirements of a friction, 405 
roller, 457 
single cone, 408 
single disc, 423 
split-ring, 449 
Velie disc, 424 

Wellman-Seaver-Morgan band, 
455 
Clutches, study of cone, 416 
of disc, 436 
of split-ring, 453 
Coefficients of friction for belting, 
159 
for friction gearing, 260 
for ball bearings, 576 
for bearings, 530 
for square threads, 107 
of linear expansion, 22 
Coil brake, 482 
Cold-rolled steel, 36 
Collar nut, 92 

thrust bearings, 543 
friction of, 550 
Tower's experiments on, 551 
work lost in, 550 
Columns, eccentric loading of, 17 

stresses in, 15 
Compensating sprocket, 257 
Compression combined with shear- 
ing, 17 
coupling, 386 
Nicholson, 388 
Compressive stress, 9 
Cone clutch, 408 

analysis of a double, 419 

of a single, 413 
asbestos fabric faced, 416 
Cadillac, 412 
engaging device for, 421 
experiments on, 418 
face angle of, 417 
Ideal multi-cone, 441 
leather faced, 416 



GENERAL INDEX 



595 



Cone clutch, motor car, 411 
National, 412 
study of, 416 
with cork inserts, 417 

face angle, 417 
Conical brakes, 472 
Connecting rod bearings, 523 
Continuous system of rope trans- 
mission, 181 
Contraction, forces due to, 2 
Copper, aluminum, 42 

zinc aluminum, 42 
Cork inserts, 407 

cone clutch with, 417 
Cotter joint, analysis of, 122 
Cotton rope transmission, 192 
Coupling bolts, 81 

clamp, 386 

Clark, 400 

compression, 386 

dimensions of clamp, 388 

flange, 383 

Francke, 396 

Hooke's, 390 

Kerr, 400 

leather-laced, 395 
-link, 394 

Nicholson compression, 388 

Nuttall, 399 

Oldham's, 390 ' 

proportions of slip, 431 

roller, 389 

rolling mill, 401 

slip, 430 
Coventry silent chain, 247 
Crane disc brakes, 478 

drum shaft, 501 

drums, design of, 208 
Critical pressure, 527 
Crown friction gearing, 267 

double, 275 

efficiency of, 275 
Crucible steel, 35 
Crushing strength of balls, 575 
Cut teeth, proportions of, 299 
Cutters for cycloidal teeth, 291 

for involute teeth, 287 
Cycloidal teeth, action of, 292 
cutters for, 291 



Cycloidal teeth, Grant's table for, 
290 
laying out, 290 
system of gearing, 288 
Cylinder heads, dast, 135 

riveted, 135 
Cylinders, thick, 130 
thin, 129 

Dedendum, 281 

Deformation due to temperature 

change, 21 
DeLaval thrust bearing, 544 
Design, principles governing, 2 
Diagram, stress-strain, 3 

of steel, 4 
Diamond tooth form for roller 

chain, 244 
Disc brakes, 473 
crane, 478 

force analysis of, 475 
clutch, analysis of, 434 
design constants for, 437 
hydraulically operated, 427 
single, 423 
study of, 436 
Dodge disc clutch, 432 
Drop-feed lubrication, 516 

-hammer, analysis of, 262 
Billings and Spencer, 262 
Toledo Machine and Tool Co., 
262 
Drums, chain, 218 

composite hoisting, 211 
conical hoisting, 209 
design of crane, 208 
wire rope, 207 
Duplex ball bearings, 574 

Eccentric loading of columns, 17 
Efficiency of boiler joints, 57 

of crown friction gearing, 275 

of manila rope transmission, 190 

of riveted connections, 55 

of spur gears, 319 

of square threads, 104 

of V-threads, 94 

worm gearing, 373 
E. G. I. clutch, 426 
Elasticity, modulus of,' 5 



596 



GENERAL INDEX 



Elastic limit, definition of, 4 
Electro-galvanizing, 46 
End connections for beams, 66 
Endurance, safe stress, 21 
Ewart chain, table of, 230 

clutch, 444 
Expansion, forces due to, 2 
Experimental conclusions of Stri- 
beck, 568 

data on hoisting tackle 179, 204 

results on friction gearing, 259 
Experiments on a cone clutch, 418 

on worm gearing, 381 

Tower's on collar thrust bear- 
ings, 551 
on step bearings, 552 

Fabroil gears, 305 
Factor of safety, 22 

table of, 23 
Factors, Lewis for stub -teeth, 298 

table of Lewis, 297 
F. and S. thrust bearings, 578 

capacities of, 581 

dimensions of, 581 
Far r el band clutch, 454 
Fastening" with eccentric loading, 101 
Fastenings, tests of belt, 155 
Feather key, 111 
Flange coupling, 383 

analysis of, 384 

marine type, 386 

proportions of, 385 
Flat key, 111 
Flexible gears, 317 

Nuttall, 318 
Flexure combined with direct stress, 
11 

stresses due to, 10 
Flooded lubrication, 518 
Forced lubrication, 518 
Forces, dead weight, 1 

due to change of velocity, 1 

due to expansion and contrac- 
tion, 2 

frictional, 1 

useful, 1 
Formulas for bearing design, 532 
Francke coupling, 396 



Frictional forces, 1 
Friction clutch, requirements of a, 
405 
coefficient of for ball bearings, 
576 

for square threads, 107 
gearing, 259 

application of spur, 261 

bearings for, 277 

bevel, 266 

coefficients of friction for, 260 

crown, 267 

double crown, 275 

efficiency of crown, 275 

experimental results on, 259 

grooved spur, 264 

plain spur, 260 
keys, 116 

of collar thrust bearings, 550 
of conical journal, 537 
of cylindrical journal, 535 
of feather keys, 117 
of pivots, 548 
spindle press, 271 

pressure developed by, 273 

Galvanizing, electro-, 46 

hot, 46 
Gear, Ingersoll slip, 317 
Nuttall flexible, 318 
Pawlings and Harnischfeger slip, 

316 
teeth, form of bevel, 323 

proportions of helical, 353 
strength of double-helical, 354 

of worm, 370 
strengthening, 311 
Gearing, application of spur friction, 
261 
bevel friction, 266 
coefficients of friction for fric- 
tion, 260 
crown friction, 267 
cycloidal system, 288 
double crown friction, 275 
efficiency of crown friction, 275 

of worm, 373 
experimental results on friction, 
259 



GENERAL INDEX 



597 



Gearing, friction, 259 

force analysis of worm, 371 
grooved spur friction, 264 
Hindley worm, 365 
involute system, 284 
load capacity of worm, 369 
materials for helical, 357 

for worm, 366 
plain spur friction, 260 
safe working stresses for, 301 
straight worm, 365 
tooth forms for worm, 367 
Gears, acute-angle bevel, 326 

advantages of double helical, 351 
applications of double helical, 

352 
arms for helical, 363 

for spur, 307 
Bakelite Micarta-D, 306 
bearing pressures due to bevel, 
338 
due to spiral bevel, 346 
circular herring bone, 364 
efficiency of spur, 319 
Fabroil, 305 
flexible, 317 
hubs for spur, 310 
large spur, 306 
materials used in, 299 
mounting bevel, 341 

helical, 363 

worm, 377 
obtuse-angle bevel, 328 
proportions of rawhide, 304 
rawhide, 302 

resultant pressure on bevel, 336 
right-angle bevel, 330 
rims for helical, 358 

for spur, 309 
skew bevel, 349 
slip, 316 
spiral bevel, 344 
strength of cast bevel, 331 

of cast spur, 294 

of cut bevel, 333 

of cut spur, 296 

of Wuest, 356 
tandem worm, 380 
tests on bevel, 348 



Gears, thrusts due to bevel, 338 
due to spiral bevel, 346 

types of helical, 350 

unequal addendum, 313 
Gib-head keys, table of, 119 
Grease lubrication, 518 
Gurney radio-thrust bearings, 583 

capacities of, 585 

dimensions of, 585 

Hangers, shaft, 526 
Hardening, 43 
Heads, cast cylinder, 135 
riveted cylinder, 135 
Heat treatments, S. A. E., 44 
Hele-Shaw clutch, 439 

analysis of, 440 
Helical gear teeth, proportions of, 353 

strength of, 354 
Helical gears, advantages of double, 
351 
applications of double, 352 
arms for, 363 
materials for, 357 
mounting, 363 
rims for, 358 
types of, 350 
Hess-Bright radial ball bearings, 576 
capacities of, 577 
dimensions of, 577 
Hoisting chain, coil, 216 
stud link, 217 
table of, 217 
drums, chain, 218 
composite, 211 
conical, 209 
wire rope, 207 
sheaves, wire rope, 204 
tackle, analysis, 178 

experimental data on, 179, 205 
wire rope, 202 
Hollow shafts, 509 
Hook tooth, 314 
Hooke's coupling, 390 

law, 3 
Horton roller clutch, 457 
Hubs for spur gears, 310 
Hunter clutch, 445 
Hyatt roller bearings, 559 



598 



GENERAL INDEX 



Hyatt roller bearings, capacities of, 
564 
dimensions of, 564 

Ideal multi-cone clutch, 441 
Involute system of gearing, 284 
teeth, action of, 287 
cutters for, 287 
Grant's table for, 286 
laying out, 285 
Iron, cast, 26 
pig, 28 
wrought, 30 

Jaw clutch, analysis of, 402 
Jeffrey-Mey-Obern chains, table of, 

232 
Johnson clutch, 451 
Joints, analysis of boiler, 55 

butt, 51 

design of boiler, 59 

efficiency of boiler, 57 

failure of, 53 

lap, 51 

splice, 70 

structural, 64 
Journals, friction of conical, 537 

friction of cylindrical, 535 

stiffness of, 534 

strength of, 534 

work lost in conical, 537 

work lost in cylindrical, 535 

Kennedy keys, 114 
Kerr coupling, 400 
Key, Barth, 113 

feather, 111 

flat, 111 

Lewis, 113 

pin, 115 

round, 115 

square, 110 

Woodruff, 111 
Keys, .dimensioning of, 120 

friction, 116 
of feather, 117 

Kennedy, 114 

on flats, 115 

strength of, 116 



Keys, table of gib-head, 119 
of round, 115 
of Woodruff, 112 

Key-seats, effect of, 511 

Knox clutch, 423 



Lap joints, 51 
Leather belting, 147 

strength of, 149 
Lenix system, 162 
Lewis factors for stub-teeth, 298 
table of, 297 
key, 113 
Limit of proportionality, 3 
Link Belt silent chains, 249 

offset connecting, 14 

spring cushioned sprockets, 257 

sprocket for, 256 

table of, 254 
Litchfield clutch, 455 
Lock nut, 90 

washer, table of, 94 
Lubrication, chain, 517 

drop-feed, 516 

flooded, 518 

forced, 518 

grease, 518 

provisions for, 516 

ring, 517 

saturated pad, 517 

system of, 516 

wick, 517 
Luder's brake, 476 



Machine bolts, 80 
screws, 85 
table of, 86 
Malleable casting, 29 
Manganese bronze, 40 
silicon steel, 38 
steel casting, 32 

applications of, 33 
Manila hoisting rope, 175 
stresses in, 176 
rope, relation between effort 
and load, 176 
sag of, 188 
selection of, 192 



GENERAL INDEX 



599 



Manila rope, transmission, efficiency 
of, 190 

force analysis of, 186 
ratio of tensions, 184 
sheave pressures for, 188 
sheaves for, 182 
transmission rope, 182 
Manufacture of shafting, 490 
Marine thrust bearings, 543 
Materials for friction clutches, 406 
for gears, 299 
for springs, 145 

table of physical properties of, 24 
Maximum normal stress theory, 495 
shear theory, 497 
strain theory, 496 
Mechanical load brakes, 476 
Medart block clutch, 445 
Merchant and Evans universal 

joint, 392 
Modulus of elasticity, 5 
of resilience, 7 
for steel, 8 
Monel metal, 41 
Moore and White clutch, 442 
Morse chain design data, 252 
silent chain, 250 
spring cushioned sprocket, 257 
Mossberg roller bearing, 556 
Mounting ball bearings, 586 

roller bearings, 564 
Multiple' system of rope transmis- 
sion, 180 

National clutch, 411 
Nickel steel, 36 

-chromium steel, 37 
Niles brake, 478 
Non-burn, 406 
Norma ball bearings, 570 

bearings, capacities of, 563 
dimensions of, 563 

roller bearing, 557 
Nuts, castellated, 93 

collar, 92 

lock, 90 

split, 93 

U. S. Standard bolts and, 78 
Nuttall coupling, 399 



Oil grooves, 516 
Oldham's coupling, 390 
Open-hearth process, 34 

Pathfinder clutch, 434 

Patch bolts, 88 

Pedestal bearings, proportions of, 

540 
Phosphor bronze, 40 
Physical properties of materials, 

table of, 24 
Pig iron, 28 

general specifications of, 29 
Pin key, 115 

plates, 73 
Pins, table of taper, 115 

taper, 124 
. Pintle bearing, 546 
Pipe thread, standard, 77 
Pitch, chordal, 281 

circular, 280 

diametral, 280 
Pivots, analysis of flat, 551 

friction of, 548 

Schiele, 553 

work lost in, 548 
Plamondon clutch, 426 
Plate and double angle connection, 
68 

and single angle connection, 64 

thickness for boiler joints, 58 
Plates, circular, 134 

elliptical, 135 

pin, 73 

rectangular, 132 

square, 133 
Poisson's ratio, 6 
Post bearings, proportions of, 540 

brake, 463 
Press, friction spindle, 271 

pressure developed by friction 
spindle, 273 
Pressures, allowable on ball bear- 
ings, 575 

bearing, 527 

critical, 527 

table of bearing, 528 
Principles governing design, 2 
Proportions of cast teeth, 295 



600 



GENERAL INDEX 



Proportions of common split bear- 
ings, 540 

of journal bearings, 539 

of pedestal bearings, 540 

of post bearings, 540 
Pulleys, cork insert, 165 

cast iron, 164 

paper, 165 

proportions of, 167 

steel, 165 

tension, 162 

tight and loose, 169 

transmitting capacity of, 166 

wood, 165 

Raceway having four-point contact. 
568 
three-point contact, 567 
two-point contact, 566 
Raceways, forms of ball, 566 
Radax ball bearing, 573 
Radial and thrust ball bearings, 573 
ball bearings, 569 
bearings having conical rollers, 
558 
having cylindrical rollers, 556 
having flexible rollers, 559 
Radiating capacity of bearings, 530 
Rawhide gears, 302 

proportions of, 304 
Raybestos, 406 

Relation between driving and driven 
sprockets, 235 
effort and load for chain, 224 
for manila rope, 176 
for wire rope, 195 
Renold tooth form, 245 
Repeating rivet group, 55 
Resilience, 6 

modulus of, 7 
for steel, 8 
Rim of spur gears, 309 
Ring lubrication, 517 
Ritter's formula, 15 
Rivet heads, forms of, 50 
' holes, 48 

recommended sizes, 59 
margin, 54 
spacing for structural work, 63 



Rivets, 48 

forms of, 49 
Rod ends, 125 
closed, 525 
open, 524 
table of B. and S., 128 

of S. A. E., 127 
Roller bearings, conical radial, 558 

data for, 562 

flexible, 559 

Mossberg, 556 

mounting of, 564 

Norma, 557 

radial, 556 

thrust, 560 

Timken, 559 
chain, 242 

Diamond tooth form for, 244 

length of, 246 

Renold tooth form for, 245 

table of Diamond, 244 

sprockets, 243 
clutch, 457 
coupling, 389 
Rolling mill coupling, 401 
Rope, bending stresses in wire, 196 
drums for wire hoisting, 207 
flat wire, 211 
manila hoisting, 175 

transmission, 182 
relation between effort and load 
for manila, 176 

between effort and load for 
wire, 195 
sag of manila, 188 

of wire, 214 
selection of manila, 192 

of wire hoisting, 202 
sheaves for wire hoisting, 204 
stresses due to slack in wire, 200 

in manila hoisting, 176 
table of wire transmission, 203 

of strengths of wire, 203 
transmission, continuous sys- 
tem, 181 

cotton, 192 

multiple system, 180 

ratio of tensions in manila, 184 
of tensions in wire, 213 



GENERAL INDEX 



601 



Rope transmission, sheaves for wire, 
213 
single loop system of wire, 
212 
wire transmission, 212 
Round keys, 115 
table of, 115 
' Rubber belting, 148 

S. A. E. heat treatments, 44 
Sag of manila rope, 188 

of wire rope, 214 
Saturated-pad lubrication, 517 
Schiele pivot, 553 

Screws, bearing pressures on power, 
108 
holding power of set, 87 
machine, 85 
set, 85 

table of cap, 84 
of machine, 86 
Sellers standard thread, 77 
Semi-steel, 30 
Set screws, 85 

holding power of, 87 
Shaft, crane dram, 501 

supporting one normal and one 

inclined load, 506 
three loads, 508 

two normal loads between bear- 
ings, 502 
with one bearing between 
the loads, 505 
Shafting, 489 

cold-rolled, 490 
commercial sizes of, 490 
design constants for, 492, 494 
drawn, 490 

effect of key-seats on, 511 
manufacture of, 490 
materials for, 489 
simple bending of, 491 

twisting of, 492 
subjected to combined twisting 
and bending, 495 
compression, 499 
torsional stiffness of, 495 

strength of, 493 
transverse stiffness of, 492 



Shafting, transverse strength of, 491 

turned, 490 
Shafts, bending moments on, 500 

hollow, 509 
Shaw brake, 481 

analysis of, 484 
Shearing combined with compres- 
sion, 17 
stress, 9 
tension, 17 
Sheave pressures for manila rope 

transmission, 188 
Sheaves, manila hoisting rope, 175 
transmission rope, 182 
plain chain, 221 
pocket chain, 222 
wire hoisting rope, 204 
transmission rope, 213 
Shererdizing, 46 
Shrouding, 311 
Silicon-manganese steel, 38 
S. K. F. ball bearing, 569 
bearings, 578 

capacities of, 579 
dimensions of, 579 
Slip coupling, 430 

proportions of, 431 
gears, 316 
Ingersoll, 317 

Pawlings and Harnischfeger, 
316 
Splice joint, 70 
Splines, integral shaft, 120 

proportions of shaft, 121 
Split bearing, 519 

nut, 93 
Split-ring clutches, 449 
analysis of, 451 
study of, 453 
Spring wire lock, 93 
Springs, concentric helical, 139 
conical, 141 
full elliptic, 145 
helical, 136 
leaf, 142 

materials for, 145 
semi-elliptic, 143 
torsion, 140 
Sprocket teeth factors, 236 



602 



GENERAL INDEX 



Sprocket tooth form, for conveyor 

chains, 236 
Sprockets, armor clad, 234 
block chain, 240 
compensating, 257 
detachable chain, 234 
Link Belt spring cushioned, 

257 
Morse spring cushioned, 257 
proportions for conveyor chain, 

238 
proportions of Link Belt silent 

chain, 256 
relation between driving and 

driven, 235 
roller chain, 243 
silent chain, 251 
spring cushioned, 256 
Spur friction gearing, 260 
applications of, 261 
grooved, 264 
gears, arms of, 307 
efficiency of, 319 
hubs of, 310 
large, 306 
rim of, 309 
strength of cast, 294 
of cut, 296 
Square thread, 77 

coefficient of friction for, 107 
Stay bolts, 89 
Steel belting, 151 
casting, 31 

manganese, 32 
chrome, 37 

chromium vanadium, 38 
cold-rolled, 36 
crucible, 35 

modulus of resilience for, 8 
nickel, 36 

-chromium, 37 
semi- 30 

silicon-manganese, 38 
stress-strain diagram of, 4 
tungsten, 38 
vanadium, 37 
Step bearings, 546 
multiple disc, 547 
single disc, 546 



Step bearings, Tower's experiments 

on, 552 
Stiffness of journals, 534 
Straight line formula for columns, 16 
Strain, definition of, 3 
Strength of journals, 534 

ultimate, 5 
Stress, compressive, 9 

definition of, 3 

safe endurance, 21 

safe working in gearing, 301 

shearing, 9 

tensile, 8 

torsional, 10 
Stresses, allowable for boiler joints, 
57 

direct combined with flexure, 11 

due to flexure, 10 

suddenly applied forces, 18 
temperature change, 22 

for leather belting, 160 

in bolts and screws, 96 

in columns, 15 

in manila hoisting ropes, 176 

in wire rope due to slack, 200 

repeated low, 20 
high, 19 

working, 22 

-strain diagram, 3 
of soft steel, 4 
Stribeck, experimental conclusions 

of, 568 
Stub teeth, 312 

dimensions of Fellows, 313 

Lewis factors for, 298 
Studs, 88 

Tackle, analysis of hoisting, 178 
experimental data on hoisting, 

179, 205 
reefed with wire rope, 202 
Taylor's experiments on belting, 161 
Teeth, dimensions of Fellows stub, 
313 
for Hindley worm gearing, 369 

worm gearing, 367 
form of bevel gear, 323 
Lewis factors for stub, 298 
proportions of cast, 295 



GENERAL INDEX 



603 



Teeth, proportions of cut, 299 
of helical gear, 353 
of worm gear, 368 
short, 311 

strength of double-helical gear, 
354 
of worm gear, 370 
strengthening gear, 311 
stub, 312 
Temperature of bearings, 533 
Tempering, 43 
Tensile stress, 8 

Tension combined with shearing, 17 
Theory, maximum normal stress, 
495 
shear, 497 
strain, 496 
Thermoid, 406 
Thread, Acme, 79 

efficiency of square, 104 

of V, 94 
forms of, 76 
Sellers standard, 77 
square, 77 
standard pipe, 77 
trapezoidal, 79 
Thrust ball bearings, 570 
Auburn, 572 
bearings, collar, 543 
DeLaval, 544 
having conical rollers, 561 
having cylindrical rollers, 560 
marine, 543 
Timken roller bearing, 559 
Toledo Machine and Tool Co. drop 

hammer, 262 
Tooth curves, 282 

forms, Diamond sprocket, 244 
conveyor chain sprocket, 236 
Renold, 245 
laying out cycloid al, 290 
involute, 285 
Torsional stress, 10 
Tower's experiments on collar thrust 
bearings, 551 
on step bearings, 552 
Tractrix, 554 
Trapezoidal thread, 79 
Tungsten steel, 38 



Union steel chain, table of, 233 

Universal joint, 390 
Bocorselski's, 391 
Merchant and Evans, 392 

Ultimate strength, 5 

Vanadium cast iron, 27 

chromium-steel, 38 

steel, 37 
V belt, block type, 171 
chain type, 172 

belting, analysis of, 172 
Vilie clutch, 424 

Washers, table of lock, 94 
Wear, adjustments for, 519 
Wellman-Seavers-Morgan clutch, 

455 
Whitney silent chain, 248 

table of, 253 
Wick lubrication, 517 
Wire hoisting rope, selection of, 202 
rope, bending stresses in, 196 
drums for, 207 
flat, 211 
relation between effort and 

load for, 195 
sag of, 214 

sheaves for hoisting, 204 
stresses due to slack, 200 
table of strengths of, 203 
transmission, ratio of ten- 
sions, 213 
sheave for, 213 
single loop system, 212 
Woodruff keys, 111 

table of, 112 
Work lost in collar thrust bearings, 
550 
conical journals, 537 
cylindrical journals, 535 
pivot, 548 
Worm, construction of, 375 

gear shaft, pressures on, 373 
gearing, efficiency of, 373 
experiments on, 381 
force analysis of, 371 
Hindley, 365 
load capacity of, 369 



604 



GENERAL INDEX 



Worm gearing, materials for, 366 Worm shaft, bearing pressures on, 



straight, 365 

strength of teeth for, 370 
teeth for Hindley, 369 
tooth forms for, 367 

gears, construction of, 375 
mounting of, 377 
proportions of, 375 
tandem, 380 

Sellers, 377 



'373 
Wrought iron, 30 
Wuest gears, strength of, 356 

Yield point, 4 
Yoke ends, 125 

table of B. and S., 128 
of S. A. E., 127 

Zinc, aluminum, 42 



INDEX TO AUTHORS, INVESTIGATORS, 

PUBLICATIONS, AND 

MANUFACTURERS 



Ahara, E. H., 191 

Akron Gear and Eng'g Co., The, 441 

Albany Hardware Specialty Mfg. 

Co., 275 
Alford, L. P., 529 
Allis-Chalmers Co., 183, 184 
American Bridge Co., 180 
American Hoist and Derick Co., 204 
American Locomotive Co., 433 
American Machinist, 87, 168, 310, 
355, 369, 415, 488, 532, 576 
American Mfg. Co., 192 
American Pulley Co., The, 165 
American Society of Civil Engineers, 

16, 180, 188 
American Society of Mechanical 
Engineers, 58, 60, 85, 89, 
90, 106, 161, 192, 259, 
260, 356 
American Society of Testing Mate- 
rials, 19 
American Steel and Wire Co., 196 
American Tool Works, The, 451 
Association of Master Steam Boiler 

Makers, 55 
Auburn Ball Bearing Co., 572 

Bach, C, 132, 135, 266, 369, 370, 381 

Barr, J. H., 145, 146 

Barth, C. G., 113, 159, 169, 301 

Bartlett, G. M., 244 

Bates, W. C, 354, 355, 356, 363 

Baush Machine Tool Co., 391 

Bearings Company of America, The, 

583 
Benjamin, C. H., 132, 168 
Billings and Spencer Co., 127, 262 
Birnie, 131 

Bonte, Prof. H., 418, 419 
Brown and Sharpe Mfg. Co., 287, 

291, 299, 367, 368 



Bruce Macbeth Engine Co., The, 394 
Bryson, 132 

Carpenter, Prof. R. C, 99 
Case Crane Co., 480 
Champion Rivet Co., 50 
Clavarino, 132 

Clyde Iron Works Co., 208, 412 
Cornell University, 531 
Wm. Cramp and Sons Ship and 
Engine Bldg. Co., 367, 428 

Day, P. C, 353, 356 

DeLaval Steam Turbine Co., 544 

Diamond Chain and Mfg. Co., 240, 

242, 243, 244, 246 
Diamond Rubber Co., The, 150 
Dodge Mfg. Co., 183, 184, 190 
Douglas, E. R., 488 

Edgar, John, 415 

Eloesser, D., 152 

Eloesser Steel Belt Co., 151 

Engineers Club of Philadelphia, 296 

Engineering Magazine, 148, 154 

Falk Co., The, 353, 356, 358, 359, 

363 
Fawcus Machine Co., 354, 355 
Fellows, E. R. 299 
Fellows Gear Shaper Co., 313 

General Electric Co., 153, 301, 305 
394, 395, 518, 529, 530 

Gleason Works, The, 314, 345, 346, 
348 

Goodenough, G. A., 218 

Goss, W. F. M., 259, 260 

Grant, G. B., 285, 286, 288, 290 

Grant, R. H., 575, 576 

Grant Gear Works, 285 

Grashof, 132, 135 



605 



606 



INDEX TO AUTHORS 



Gratbn and Knight Mfg. Co., 172 
Greaves Klusman Tool Co., 451 
Griffin, C. L., 501 
Guest, Prof. 497 

Gurney Ball Bearing Co., 343, 573, 
583, 584, 585 

Halsey, F. A., 238 

Hancock, Prof., 499 

Hele-Shaw, Prof., 439, 440 

Hess-Bright Mfg. Co., 576, 577, 578 

Hewitt, W., 214 

Hindley, 365, 366, 369, 376 

Hooke, 3, 201 

Hunt, C. W., 295, 299, 311 

Hyatt Roller Bearing Co., 559, 564 

Illinois Steel Co., 430 
Illmer, Louis, 530, 531, 532 
Ingersoll Milling Machine Co., 317, 

521 
Institution of Mechanical Engineers, 

551 
Jeffrey Mfg. Co., 232, 236 
Johns-Manville Co., 406 
Johnson, Herman, 310 
Johnson, Thos. H., 16 

Kelso, 531 
Kenyon, E., 193 
Kerr, C. V., 400 

Keystone-Hindley Gear Co., 369 
Kimball, D. C, 145, 146 
Kingsbury, Prof. A., 106, 107 
Knight, William, 532 

Lame, 130 

Lanchester, 369, 376 

Lasche, 530, 531 

Lewis, Wilford, 113, 296, 297, 298, 

301, 311, 333, 355 
Link Belt Co., 229, 232, 237, 249, 251 
Litchfield Foundry and Machine 

Co., 455 
Logue, C. H., 299, 313 
Lucas Machine Tool Co., 410 

Machinery, 348 
Marx, Prof. G. H., 527 



Maurer, Prof., 531 

Mechanical Engineers' Handbook, 

138, 141 
Merchant and Evans Co., 392 
Merriman, Mansfield, 16, 17, 132 
Mesta Machine Co., 306 
Metten, J. F., 428 
Miller, Spencer, 187 
Mitchell, S. P., 180 
Moore, H. F., 19, 21, 511, 527 
Moore, L. E., 218 
Morrison, C. J., 148, 154, 160 
Morrison, E. R., 143 
Morse Chain Co., 250, 251, 252, 257 

National Association of Cotton Mfr., 

166 
New Departure Mfg. Co., The, 343, 

379, 573, 574 
New Process Rawhide Co., The, 

300, 304 
Niagara Falls Power Co., 561 
Nichols, Prof., 488 
Nicholson and Co., W. H., 389 
Niles-Bement-Pond Co., 478, 482 
Nordberg Mfg. Co., 115, 464. 
Norma Company of America, The, 

557, 562, 570 
Nuttall Co., The R. D., 317, 364, 399 

Pawlings and Harnischfeger Co., 

316, 479, 547 
Pederson, Axel K, 518, 530 
Philosophical Magazine, 497 
Pinkney, B. H. D., 87 
Poisson, 6, 497 ■ 
Power, 530 

Rankine, Prof., 495 
Renold, Hans, 245, 249 
Ritter, 15, 17 
Roser, E., 369, 370, 381 

Saint Venant, 496 

Sawdon, W. M., 166 

Seeley, F. B., 19, 21 

Sellers Co., William, 377 

Shaw Electric Crane Co., 481 

S. K. F. Bearing Co., 569, 578, 579 



INDEX TO AUTHORS 



607 



Smith, A. W., 527 

Smith, C. A. M., 499 

Smith, L. G., 299 

Society of Automobile Engineers, 

41, 42, 43, 44, 82, 93, 121, 

127, 417, 576 
South Wales Institute of Engineers, 

193 
Standard Machinery Co., 557 
Stephens-Adamson Mfg. Co., 540 
Stribeck, Prof., 381, 527, 562, 568, 

575, 576 

Taylor, F. W., 161 

Thomas, C, 531 

Timken Roller Bearing Co., 379, 559 

Toledo Machine and Tool Co., 262 



Tower, Beauchamp, 531, 551, 552 
Trenton Iron Co., 214 

Union Chain and Mfg. Co., 233, 235 
University of Illinois Experiment 

Station, 218, 511 
University of Missouri, 532 

Weisbach, 318 

Wellman-Seavers-Morgan Co., 455 
Westcott, A. L., 532 
Westinghouse Electric and Mfg.Co., 

301, 386 
Whitney Mfg. Co., Ill, 248, 253 

Zeh and Hahnemann Co., 271 
Zeitschrift des Vereins deutcher 
Ingenieure, 369, 418 









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